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s. Unsourced material may be challenged and removed. (May 2011) (Learn how and when to remove this template message) [imagelink] Waste management in Kathmandu, Nepal [imagelink] Waste management in Stockholm, Sweden <span>Waste management or Waste disposal is all the activities and actions required to manage waste from its inception to its final disposal. [1] This includes amongst other things, collection, transport, treatment and disposal of waste together with monitoring and regulation. It also encompasses the legal and regulatory framework that relates to waste management encompassing guidance on recycling etc. The term normally relates to all kinds of waste, whether generated during the extraction of raw materials, the processing of raw materials into intermediate and final products, the co

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An autoenco der is the com bination of an enco der function that con v erts the input data into a diﬀeren t representation, and a deco der function that conv erts the new representation back into the o

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An autoenco der is the com bination of an enco der function that con v erts the input data into a diﬀeren t representation, and a deco der function that conv erts the new representation back into the original format

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An autoenco der is the com bination of an enco der function that con v erts the input data into a diﬀeren t representation, and a deco der function that conv erts the new representation back into the original format

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det determinant.

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det determinant.

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eig eigenvalue decomposition

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eig eigenvalue decomposition

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svd singular value decomposition

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svd singular value decomposition

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The piecing together of the complete glycolytic pathway in the 1930s was a major triumph of biochemistry

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The piecing together of the complete glycolytic pathway in the 1930s was a major triumph of biochemistry

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Two central reactions in glycolysis (steps 6 and 7) convert the three-carbon sugar inter- mediate glyceraldehyde 3-phosphate (an aldehyde) into 3-phosphoglycerate (a carboxylic acid; see Panel 2–8, pp. 104–105), thus oxidizing an aldehyde group to a carboxylic acid group.

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Two central reactions in glycolysis (steps 6 and 7) convert the three-carbon sugar inter- mediate glyceraldehyde 3-phosphate (an aldehyde) into 3-phosphoglycerate (a carboxylic acid; see Panel 2–8, pp. 104–105), thus oxidizing an aldehyde group to a carboxylic acid group.

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In step 6, the enzyme glyceraldehyde 3-phosphate dehydrogenase couples the energetically favorable oxidation of an aldehyde to the energetically unfavorable formation of a high-energy phosphate bond. At the same time, it enables energy to be s

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In step 6, the enzyme glyceraldehyde 3-phosphate dehydrogenase couples the energetically favorable oxidation of an aldehyde to the energetically unfavorable formation of a high-energy phosphate bond. At the same time, it enables energy to be stored in NADH. The formation of the high-energy phosphate bond is driven by the oxidation reaction, and the enzyme thereby acts like the

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W e use the − sign to index the complement of a set. F or example x − 1 is the v ector con taining all elemen ts of x except for x 1 , and x − S is the v ector con taining all of the elements of exce

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W e can identi fy all of the n um b ers with v ertical co ordinate i b y writing a “ ” for the horizontal : co ordinate. F or example, A i, : denotes the horizon tal cross section of A with v ertical co ordinate i

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The transp ose of a matrix is the mirror image of the matrix across a diagonal line, called the main diagonal , running do wn and to the righ t, starting from its upp er left corner.

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Note that the standard pro duct of tw o matrices is just a matrix con taining not the pro duct of the individual elements. Suc h an op eration exists and is called the elemen t-wise pro duct Hadamard pro duct or , and is denoted as . A

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the dot pro duct b etw een tw o v ectors is comm utativ e: x y y = x

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The transp ose of a matrix pro duct has a simple form: \((AB)^{T} = B^{T}A^{T}\) . (2.9) This allo ws us to demonstrate equation , b y exploiting the fact that the v alue 2.8 of suc h a pro duct is a scalar and therefore equal to its o wn transp ose:\( x^Ty = (x^Ty)^T

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W e can now solve equation Ax = b by the following steps: 1. A -1 Ax = A −1 b 2. I n x = A −1 b

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fatty acids enter the bloodstream, where they bind to the abundant blood protein, serum albumin.

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In addition to three molecules of NADH, each turn of the cycle also produces one molecule of FADH 2 (reduced flavin adenine dinucleotide) from FAD (see Figure 2–39), and one mol- ecule of the ribonucleoside triphosphate GTP from GDP

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In addition to three molecules of NADH, each turn of the cycle also produces one molecule of FADH 2 (reduced flavin adenine dinucleotide) from FAD (see Figure 2–39), and one mol- ecule of the ribonucleoside triphosphate GTP from GDP

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n total, the complete oxidation of a molecule of glucose to H 2 O and CO 2 is used by the cell to produce about 30 molecules of ATP.

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pan>All of the nitrogens in the purine and pyrimidine bases (as well as some of the carbons) are derived from the plentiful amino acids glutamine, aspartic acid, and glycine, whereas the ribose and deoxyribose sugars are derived from glucose.<span><body><html>

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All of the nitrogens in the purine and pyrimidine bases (as well as some of the carbons) are derived from the plentiful amino acids glutamine, aspartic acid, and glycine, whereas the ribose and deoxyribose sugars are derived from glucose.

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All of the nitrogens in the purine and pyrimidine bases (as well as some of the carbons) are derived from the plentiful amino acids glutamine, aspartic acid, and glycine, whereas the ribose and deoxyribose sugars are derived from glucose.

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All of the nitrogens in the purine and pyrimidine bases (as well as some of the carbons) are derived from the plentiful amino acids glutamine, aspartic acid, and glycine, whereas the ribose and deoxyribose sugars are derived from glucose.<

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Sulfur is abundant on Earth in its most oxidized form, sulfate (SO 4 2– ). To be useful for life, sulfate must be reduced to sulfide (S 2– ), the oxidation state of sulfur required for the synthesis of essential biological molecules, including the amino ac

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Sulfur is abundant on Earth in its most oxidized form, sulfate (SO 4 2– ). To be useful for life, sulfate must be reduced to sulfide (S 2– ), the oxidation state of sulfur required for the synthesis of essential biological molecules, including the amino acids methionine and cysteine, coenzyme A (see Figure 2–39), and the i

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is abundant on Earth in its most oxidized form, sulfate (SO 4 2– ). To be useful for life, sulfate must be reduced to sulfide (S 2– ), the oxidation state of sulfur required for the synthesis of essential biological molecules, including the <span>amino acids methionine and cysteine, coenzyme A (see Figure 2–39), and the iron-sulfur centers essential for electron transport (see Figure 14–16)<span><body><html>

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Humans and other animals cannot reduce sulfate and must therefore acquire the sulfur they need for their metabolism in the food that they eat

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Humans and other animals cannot reduce sulfate and must therefore acquire the sulfur they need for their metabolism in the food that they eat

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The characteristic “size” for each atom is specified by a unique van der Waals radius. The contact distance between any two noncovalently bonded atoms is the sum of their van der Waals radii

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The characteristic “size” for each atom is specified by a unique van der Waals radius. The contact distance between any two noncovalently bonded atoms is the sum of their van der Waals radii

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The characteristic “size” for each atom is specified by a unique van der Waals radius. The contact distance between any two noncovalently bonded atoms is the sum of their van der Waals radii

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Reshaping can also be done with the reshape function

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Reshaping can also be done with the reshape function

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On the right-hand side of an assignment, a(:) gives all the elements of a strung out by columns in one long column vector

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On the right-hand side of an assignment, a(:) gives all the elements of a strung out by columns in one long column vector

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As a special case, a single colon subscript may be used to replace all the elements of a matrix with a scalar, e.g., a(:) = -1

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As a special case, a single colon subscript may be used to replace all the elements of a matrix with a scalar, e.g., a(:) = -1

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You can’t delete a single element from a matrix while keeping it a matrix, so a statement like a(1,2) = [ ] results in an error

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T o analyze ho w man y solutions the equation has, we can think of the columns of A as sp ecifying diﬀerent directions we can tra v el from the origin (the p oin t sp eciﬁed b y the v ector of all zeros), and determine ho w many wa ys there are of reac hing b . In this view, each element of x sp eciﬁes ho w far we should trav el in ea

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we can think of the columns of A as sp ecifying diﬀerent directions we can tra v el from the origin (the p oin t sp eciﬁed b y the v ector of all zeros), and determine ho w many wa ys there are of reac hing b . In this view, each element of x <span>sp eciﬁes ho w far we should trav el in eac h of these directions, with x i sp ecifying how far to mo v e in the direction of column :<span><body><html>

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ra v el from the origin (the p oin t sp eciﬁed b y the v ector of all zeros), and determine ho w many wa ys there are of reac hing b . In this view, each element of x sp eciﬁes ho w far we should trav el in eac h of these directions, with x i <span>sp ecifying how far to mo v e in the direction of column :<span><body><html>

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The “evidence” for the model, p(D), is the overall probability of the data according to the model, determined by averaging across all possible parameter values weighted by the strength of belief in those parameter value

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The “evidence” for the model, p(D), is the overall probability of the data according to the model, determined by averaging across all possible parameter values weighted by the strength of belief in th

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body>In many models, the probability of data, p(D|θ), does not depend in any way on other data. That is, the joint probability p(D, D |θ) equals p(D|θ)·p(D |θ). In other words, in this sort of model, the data probabilities are independent (recall that independence was defined in Section 4.4.2). Under this condition, then the order of updating has no effect of the final posterior<body><html>

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a general phenomenon in Bayesian inference: The posterior is a compromise between the prior distribution and the likelihood function.

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The compromise favors the prior to the extent that the prior distribution is sharply peaked and the data are few. The compromise favors the likelihood function (i.e., the data) to the extent that the prior distribution is flat and the data are many.

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The compromise favors the prior to the extent that the prior distribution is sharply peaked and the data are few. The compromise favors the likelihood function (i.e., the data) to the extent that the prior distribution is flat and the data are many.

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The compromise favors the prior to the extent that the prior distribution is sharply peaked and the data are few. The compromise favors the likelihood function (i.e., the data) to the extent that the prior distributio

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The compromise favors the prior to the extent that the prior distribution is sharply peaked and the data are few. The compromise favors the likelihood function (i.e., the data) to the extent that the prior distribution is flat and the data are many.

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the mode of the posterior distribution is between the mode of the prior distribution and the mode of the likelihood function, but the posterior mode is closer to the likelihood mode for larger sample sizes

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the mode of the posterior distribution is between the mode of the prior distribution and the mode of the likelihood function, but the posterior mode is closer to the likelihood mode for larger sample sizes

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the functions zeros, ones and rand generate matrices of 1s, 0s and random numbers, respectively. With a single argument n, they generate n × n (square) matrices. With two arguments n and m they generate n × m ma- trices.

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The function eye(n) generates an n × n identity matrix, i.e., a matrix with 1s on the main ‘diagonal’, and 0s everywhere else

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diag extracts or creates a diagonal

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diag extracts or creates a diagonal

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triu extracts the upper triangular part

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triu extracts the upper triangular part

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although the overall conformation of each pro- tein is unique, two regular folding patterns are often found within them. Both pat- terns were discovered more than 60 years ago from studies of hair and silk

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although the overall conformation of each pro- tein is unique, two regular folding patterns are often found within them. Both pat- terns were discovered more than 60 years ago from studies of hair and silk

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Within a year of the discovery of the α helix, a second folded structure, called a β sheet, was found in the protein fibroin, the major constituent of silk. These two patterns are particularly comm

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Within a year of the discovery of the α helix, a second folded structure, called a β sheet, was found in the protein fibroin, the major constituent of silk. These two patterns are particularly common because they result from hydro- gen-bonding between the N–H and C=O groups

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Within a year of the discovery of the α helix, a second folded structure, called a β sheet, was found in the protein fibroin, the major constituent of silk. These two patterns are particularly common because they result from hydro- gen-bonding between the N–H and C=O groups in the polypeptide backbone, with

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Within a year of the discovery of the α helix, a second folded structure, called a β sheet, was found in the protein fibroin, the major constituent of silk. These two patterns are particularly common because they result from hydro- gen-bonding between the N–H and C=O groups in the polypeptide backbone, without involving the side chains o

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l>Within a year of the discovery of the α helix, a second folded structure, called a β sheet, was found in the protein fibroin, the major constituent of silk. These two patterns are particularly common because they result from hydro- gen-bonding between the N–H and C=O groups in the polypeptide backbone, without involving the side chains of the amino acids<html>

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inv inverse.

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inv inverse.

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qr orthogonal factorization

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qr orthogonal factorization

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body>To see how MATLAB solves this system, first recall that the left division operator \ may be used on scalars, i.e., a\bis the same as b/aif a and b are scalars. However, it can also be used on vectors and matrices, in order to solve linear equations<body><html>

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When we have more equations than unknowns, the system is called over- determined

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When we have more equations than unknowns, the system is called over- determined

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The residual r=A*x-b is now r= -0.3333 -0.3333 0.3333 What happens in this case is that MATLAB produces the least squares best fit. This is the value of x which makes the magnitude of r, i.e., r(1) 2 + r(2) 2 + r(3) 2 , as small as possible.

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The residual r=A*x-b is now r= -0.3333 -0.3333 0.3333 What happens in this case is that MATLAB produces the least squares best fit. This is the value of x which makes the magnitude of r, i.e., r(1) 2 + r(2) 2 + r(3) 2 , as small as possible.

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If there are fewer equations than unknowns, the system is called under- determined. In this case there are an infinite number of solutions; MATLAB will find one which has zeros for some of the unknowns.

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If there are fewer equations than unknowns, the system is called under- determined. In this case there are an infinite number of solutions; MATLAB will find one which has zeros for some of the unknowns.

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If there are fewer equations than unknowns, the system is called under- determined. In this case there are an infinite number of solutions; MATLAB will find one which has zeros for some of the unknowns.

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A system like this is called ill-conditioned, meaning that a small change in the co- efficients leads to a large change in the solution. The MATLAB function rcond returns the condition estimator, which tests for ill conditioning</s

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A system like this is called ill-conditioned, meaning that a small change in the co- efficients leads to a large change in the solution. The MATLAB function rcond returns the condition estimator, which tests for ill conditioning

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A system like this is called ill-conditioned, meaning that a small change in the co- efficients leads to a large change in the solution. The MATLAB function rcond returns the condition estimator, which tests for ill conditioning

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A system like this is called ill-conditioned, meaning that a small change in the co- efficients leads to a large change in the solution. The MATLAB function rcond returns the condition estimator, which tests for ill conditioning

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The function spy provides a neat visualization of sparse matrices.

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The function spy provides a neat visualization of sparse matrices.

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The space of possibilities is the joint parameter space involving all combinations of parameter values. If we represent each parameter with a comb of, say, 1,000 values, then for P parameters there are 1,000 P combinations of parameter values.

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The space of possibilities is the joint parameter space involving all combinations of parameter values. If we represent each parameter with a comb of, say, 1,000 values, then for P parameters there are 1,000 P combinations of parameter values.

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Note that all of the N–H groups point up in this diagram and that all of the C=O groups point down (toward the C-terminus); this gives a polarity to the helix, with the C-terminus having a partial negative and the N-terminus a partial positive charge (

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Note that all of the N–H groups point up in this diagram and that all of the C=O groups point down (toward the C-terminus); this gives a polarity to the helix, with the C-terminus having a partial negative and the N-terminus a partial positive charge (

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y>Note that all of the N–H groups point up in this diagram and that all of the C=O groups point down (toward the C-terminus); this gives a polarity to the helix, with the C-terminus having a partial negative and the N-terminus a partial positive charge (<body><html>

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The β sheet is shown in (C) and (D). In this example, adjacent peptide chains run in opposite (antiparallel) directions. Hydrogen-bonding between peptide bonds in different strands holds the individual polypeptide chains (strands) together in a β sheet, and the amino acid side chains in each

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nt peptide chains run in opposite (antiparallel) directions. Hydrogen-bonding between peptide bonds in different strands holds the individual polypeptide chains (strands) together in a β sheet, and the amino acid side chains in each strand <span>alternately project above and below the plane of the sheet<span><body><html>

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The β sheet is shown in (C) and (D). In this example, adjacent peptide chains run in opposite (antiparallel) directions. Hydrogen-bonding between peptide bonds in different strands holds the individual polypeptide chains (strands) together in a β sheet, and the amino acid side chains in each strand alternately project a

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An α helix is generated when a single polypeptide chain twists around on itself to form a rigid cylinder. A hydrogen bond forms between every fourth peptide bond, linking the C=O of one peptide

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An α helix is generated when a single polypeptide chain twists around on itself to form a rigid cylinder. A hydrogen bond forms between every fourth peptide bond, linking the C=O of one peptide bond to the N–H of another (see Figure 3–7A). This gives rise to a regular helix with a complete turn every 3.6 amino acids.

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around on itself to form a rigid cylinder. A hydrogen bond forms between every fourth peptide bond, linking the C=O of one peptide bond to the N–H of another (see Figure 3–7A). This gives rise to a regular helix with a complete turn every <span>3.6 amino acids.<span><body><html>

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α helices wrap around each other to form a particularly sta- ble structure, known as a coiled-coil. This structure can form when the two (or in some cases, three or four) α helices have most of their nonpolar (hydrophobic) side chains on one side, so that they can twist around each

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α helices wrap around each other to form a particularly sta- ble structure, known as a coiled-coil. This structure can form when the two (or in some cases, three or four) α helices have most of

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α helices wrap around each other to form a particularly sta- ble structure, known as a coiled-coil. This structure can form when the two (or in some cases, three or four) α helices have most of their nonpolar (hydrophobic) side chains on one side, so that they can twist around each other with these side chains facing inward

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This is the protein domain, a substructure produced by any contiguous part of a polypeptide chain that can fold independently of the rest of the protein into a compact, stable structure.

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This is the protein domain, a substructure produced by any contiguous part of a polypeptide chain that can fold independently of the rest of the protein into a compact, stable structure. </

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A domain usually contains between 40 and 350 amino acids, and it is the modular unit from which many larger proteins are constructed

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the Src protein kinase, which functions in signaling pathways inside vertebrate cells

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the Src protein kinase, which functions in signaling pathways inside vertebrate cells

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The smallest protein molecules contain only a single domain, whereas larger proteins can contain several dozen domains, often connected to each other by short, relatively unstructured lengths of polypeptide chain that can act as flexibl

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The smallest protein molecules contain only a single domain, whereas larger proteins can contain several dozen domains, often connected to each other by short, relatively unstructured lengths of polypeptide chain that can act as flexible hinges between domains.

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The smallest protein molecules contain only a single domain, whereas larger proteins can contain several dozen domains, often connected to each other by short, relatively unstructured lengths of polypeptide chain that can act as flexible hinges between domains.

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Since each of the 20 amino acids is chemically distinct and each can, in princi- ple, occur at any position in a protein chain, there are 20 × 20 × 20 × 20 = 160,000 different possible polypeptide chains four amino acids long, or 20 n different pos- sible polypeptide chains n amino acids long. For a typical protein length of about 300 amino acids

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20 = 160,000 different possible polypeptide chains four amino acids long, or 20 n different pos- sible polypeptide chains n amino acids long. For a typical protein length of about 300 amino acids, a cell could theoretically make more than <span>10 390 (20 300 ) different polypeptide chains<span><body><html>

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Many similar examples show that two pro- teins with more than 25% identity in their amino acid sequences usually share the same overall structure

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Many similar examples show that two pro- teins with more than 25% identity in their amino acid sequences usually share the same overall structure

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there are a limited number of ways in which protein domains fold up in nature—maybe as few as 2000, if we consider all organisms. For most of these so-called protein folds, representative structures have been determined.

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The present database of known protein sequences contains more than twenty million entries

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The encoded polypeptides range widely in size, from 6 amino acids to a gigantic pro- tein of 33,000 amino acids.

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The encoded polypeptides range widely in size, from 6 amino acids to a gigantic pro- tein of 33,000 amino acids.

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dy>most proteins are composed of a series of protein domains, in which different regions of the polypeptide chain fold independently to form compact structures. Such multidomain proteins are believed to have originated from the accidental joining of the DNA sequences that encode each domain, cre- ating a new gene.<body><html>

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In an evolutionary process called domain shuffling, many large proteins have evolved through the joining of preexisting domains in new com- binations (Figure 3–14). Novel binding surfaces have often been created at the juxtaposition o

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β-sheet-based domains may have achieved their evolutionary success because they provide a convenient framework for the generation of new binding sites for ligands, requiring only small changes to their protruding loops

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Stiff extended structures composed of a series of domains are especially common in extracellular matrix molecules and in the extracellular portions of cell-surface receptor proteins.

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Stiff extended structures composed of a series of domains are especially common in extracellular matrix molecules and in the extracellular portions of cell-surface receptor proteins.

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Stiff extended structures composed of a series of domains are especially common in extracellular matrix molecules and in the extracellular portions of cell-surface receptor proteins.

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N- and C-terminal ends at opposite poles of the domain. When the DNA encoding such a domain undergoes tandem duplica- tion, which is not unusual in the evolution of genomes (discussed in Chapter 4), the duplicated domains with this “in-line” arrangement can be readily linked in series to form extended str

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half of all domain families are common to archaea, bacteria, and eukaryotes, only about 5% of the two-domain combi- nations are similarly shared.

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half of all domain families are common to archaea, bacteria, and eukaryotes, only about 5% of the two-domain combi- nations are similarly shared.

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most proteins containing especially useful two-domain combinations arose through domain shuffling rel- atively late in evolution

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most proteins containing especially useful two-domain combinations arose through domain shuffling rel- atively late in evolution

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we currently lack even the tiniest hint of what the function might be for more than 10,000 of the proteins that have thus far been identified through examining the human genome

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combinations of protein domains, with the result that there are nearly twice as many combinations of domains found in human proteins as in a worm or a fly.

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Determining whether Ax = b has a solution th us amounts to testing whether b is in the span of the columns of A . This particular span is known as the column space range or the of . A

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Determining whether Ax = b has a solution th us amounts to testing whether b is in the span of the columns of A . This particular span is known as the column space range or the of . A

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Ha ving n m ≥ is only a necessary condition for ev ery p oin t to ha ve a solution. It is not a suﬃcien t condition, b ecause it is p ossible for some of the columns to b e redundan t.

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Ha ving n m ≥ is only a necessary condition for ev ery p oin t to ha ve a solution. It is not a suﬃcien t condition, b ecause it is p ossible for some of the columns to b e redundan t.<

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Ha ving n m ≥ is only a necessary condition for ev ery p oin t to ha ve a solution. It is not a suﬃcien t condition, b ecause it is p ossible for some of the columns to b e redundan t.

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

F ormally , this kind of redundancy is kno wn as linear dep endence . A set of v ectors is linearly indep enden t if no v ector in the set is a linear combination of the other vectors.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

F ormally , this kind of redundancy is kno wn as linear dep endence . A set of v ectors is linearly indep enden t if no v ector in the set is a linear combination of the other vectors.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

the deriv ativ es of the squared L 2 norm with resp ect to each element of x eac h dep end only on the corresp onding elemen t of x , while all of the deriv ativ es of the L 2 norm dep end on the en tire vector</b

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

the deriv ativ es of the squared L 2 norm with resp ect to each element of x eac h dep end only on the corresp onding elemen t of x , while all of the deriv ativ es of the L 2 norm dep end on the en tire vector

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

the deriv ativ es of the squared L 2 norm with resp ect to each element of x eac h dep end only on the corresp onding elemen t of x , while all of the deriv ativ es of the L 2 norm dep end on the en tire vector

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The L 1 norm is commonly used in machine learning when the diﬀerence b etw een zero and nonzero elements is v ery imp ortan t. Every time an element of x mo v es a w a y from 0 b y , the L 1 norm

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The L 1 norm is commonly used in machine learning when the diﬀerence b etw een zero and nonzero elements is v ery imp ortan t. Every time an element of x mo v es a w a y from 0 b y , the L 1 norm increases b y .

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Diagonal matrices are of interest in part b ecause multiplying by a diagonal matrix is very computationally eﬃcien t.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Diagonal matrices are of interest in part b ecause multiplying by a diagonal matrix is very computationally eﬃcien t.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

T o compute diag(v)x , we only need to scale each element x i b y v i . In other w ords, diag(v)x = x⋅yx⋅y

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

T o compute diag(v)x , we only need to scale each element x i b y v i . In other w ords, diag(v)x = x⋅yx⋅y

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Non-square diagonal matrices do not hav e inv erses but it is still p ossible to multiply by them cheaply

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Non-square diagonal matrices do not hav e inv erses but it is still p ossible to multiply by them cheaply

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Non-square diagonal matrices do not hav e inv erses but it is still p ossible to multiply by them cheaply

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A unit vector is a vector with unit norm ||x|| 2 = 1

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A unit vector is a vector with unit norm ||x|| 2 = 1

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A vector x and a vector y are orthogonal to each other if x y = 0 .

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A vector x and a vector y are orthogonal to each other if x T y = 0 .

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In R n , at most n v ectors ma y b e mutually orthogonal with nonzero norm. If the v ectors are not only orthogonal but also ha ve unit norm, we call them orthonormal

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In R n , at most n v ectors ma y b e mutually orthogonal with nonzero norm. If the v ectors are not only orthogonal but also ha ve unit norm, we call them orthonormal