Chapter 5: Matrices - University Of New South Wales
Chapter 5: MatricesDaniel ChanUNSWTerm 1 2021Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20211 / 33
In this chapterDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
In this chapterMatrices were first introduced in the Chinese “Nine Chapters on theMathematical Art” to solve linear eqns.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
In this chapterMatrices were first introduced in the Chinese “Nine Chapters on theMathematical Art” to solve linear eqns.In the mid-1800s, senior wrangler Arthur Cayley studied matrices in their ownright and showed how they have an interesting and useful algebra associatedto them.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
In this chapterMatrices were first introduced in the Chinese “Nine Chapters on theMathematical Art” to solve linear eqns.In the mid-1800s, senior wrangler Arthur Cayley studied matrices in their ownright and showed how they have an interesting and useful algebra associatedto them.We will look at Cayley’s ideas and extend vector arithmetic to matrices andeven show there is matrix multiplication akin to multiplying numbers.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
In this chapterMatrices were first introduced in the Chinese “Nine Chapters on theMathematical Art” to solve linear eqns.In the mid-1800s, senior wrangler Arthur Cayley studied matrices in their ownright and showed how they have an interesting and useful algebra associatedto them.We will look at Cayley’s ideas and extend vector arithmetic to matrices andeven show there is matrix multiplication akin to multiplying numbers.These ideas will not only shed light on solving linear eqns,Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
In this chapterMatrices were first introduced in the Chinese “Nine Chapters on theMathematical Art” to solve linear eqns.In the mid-1800s, senior wrangler Arthur Cayley studied matrices in their ownright and showed how they have an interesting and useful algebra associatedto them.We will look at Cayley’s ideas and extend vector arithmetic to matrices andeven show there is matrix multiplication akin to multiplying numbers.These ideas will not only shed light on solving linear eqns, they will also beuseful later when you look at multivariable functions and mappings.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20212 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnNotationWe abbreviate the above to A (aij ) and call aij the ij-th entry of A.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnNotationWe abbreviate the above to A (aij ) and call aij the ij-th entry of A.Also write [A]ij for aij .Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnNotationWe abbreviate the above to A (aij ) and call aij the ij-th entry of A.Also write [A]ij for aij .We say the size of A is m n because it hasDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnNotationWe abbreviate the above to A (aij ) and call aij the ij-th entry of A.Also write [A]ij for aij .We say the size of A is m n because it hasMmn (R) (resp Mmn (C)) denote the set of all m n-matrices with real entries(resp complex entries). Sometimes abbreviate to Mmn if the scalars areunderstood or irrelevant.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Some new notation for matricesRecall an m n-matrix is an array of (for us) scalars (real or complex). a11 a12 · · · a1n a21 a22 · · · a2n A . . . am1 am2 · · · amnNotationWe abbreviate the above to A (aij ) and call aij the ij-th entry of A.Also write [A]ij for aij .We say the size of A is m n because it hasMmn (R) (resp Mmn (C)) denote the set of all m n-matrices with real entries(resp complex entries). Sometimes abbreviate to Mmn if the scalars areunderstood or irrelevant.E.g. A length m column vector is anDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20213 / 33
Revise matrix-vector productDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . ThenDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnAlternatively, the i-th entry of Ax isDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnAlternatively, the i-th entry of Ax is[Ax]i ai1 x1 . . . ain xn (ai1Daniel Chan (UNSW)Chapter 5: Matrices x1 x2 . . . ain ) . . . xnTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnAlternatively, the i-th entry of Ax is[Ax]i ai1 x1 . . . ain xn (ai1 x1 x2 . . . ain ) . . . xnNote similarity with dot products.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnAlternatively, the i-th entry of Ax is[Ax]i ai1 x1 . . . ain xn (ai1 x1 x2 . . . ain ) . . . xnNote similarity with dot products.A induces the linear function T : Rn Rm : x 7 Ax.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Revise matrix-vector productLet A (aij ) (a1 a2 . . . an ) Mmn . Then x1 x2 A . . x1 a1 x2 a2 . . . xn an .xnAlternatively, the i-th entry of Ax is[Ax]i ai1 x1 . . . ain xn (ai1 x1 x2 . . . ain ) . . . xnNote similarity with dot products.A induces the linear function T : Rn Rm : x 7 Ax.Note We will write all our results for matrices with real entries, but there areobvious analogues over the complexes.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20214 / 33
Arithmetic of matricesDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 1 12Daniel Chan (UNSW)4 1 6 5Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 4 6 1 12 1 5 1 2 37 0 1 1Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 4 6 1 12 1 5 1 2 37 0 1 1In formulasMatrix arithmeticFor A, B Mmn (R), λ R, the entries of A B, λA Mmn (R) areDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 4 6 1 12 1 5 1 2 37 0 1 1In formulasMatrix arithmeticFor A, B Mmn (R), λ R, the entries of A B, λA Mmn (R) are[A B]ij [A]ij [B]ijDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 4 6 1 12 1 5 1 2 37 0 1 1In formulasMatrix arithmeticFor A, B Mmn (R), λ R, the entries of A B, λA Mmn (R) are[A B]ij [A]ij [B]ij[λA]ij λ[A]ijDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Arithmetic of matricesJust as for vectors, we can define matrix addition and scalar multiplication to beentry-wise addition and scalar multiplicationE.g. 10 2 33 4 6 1 12 1 5 1 2 37 0 1 1In formulasMatrix arithmeticFor A, B Mmn (R), λ R, the entries of A B, λA Mmn (R) are[A B]ij [A]ij [B]ij[λA]ij λ[A]ijN.B. We don’t define the sum of matrices of different sizes (just as is the case forvectors).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20215 / 33
Linear combinations and subtractionDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 10Daniel Chan (UNSW)21 3 1Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 1021 3 1DefinitionThe zero matrix 0 has all entries 0. (There’s one for each size m n.)Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 1021 3 1DefinitionThe zero matrix 0 has all entries 0. (There’s one for each size m n.)A 0 Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 1021 3 1DefinitionThe zero matrix 0 has all entries 0. (There’s one for each size m n.)A 0 The negative of A Mmn is A : ( 1)A.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 1021 3 1DefinitionThe zero matrix 0 has all entries 0. (There’s one for each size m n.)A 0 The negative of A Mmn is A : ( 1)A. Hence A ( A) Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Linear combinations and subtractionE.g. We can also form linear combinations of matrices 1021 31 ( 1) 1021 3 1DefinitionThe zero matrix 0 has all entries 0. (There’s one for each size m n.)A 0 The negative of A Mmn is A : ( 1)A. Hence A ( A) The difference A B A ( B) if A, B have the same size.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20216 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x RnDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.(λA)x λ(Ax).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.(λA)x λ(Ax).Proof. Suppose n 2 (else need more space) so A (a1 a2 ), B Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.(λA)x λ(Ax).Proof. Suppose n 2 (else need more space) so A (a1 a2 ), B Upshot Recall that in calculus, you define the sum and scalar multiple offunctions pointwise, (f g )(x) f (x) g (x), (λf )(x) λf (x).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.(λA)x λ(Ax).Proof. Suppose n 2 (else need more space) so A (a1 a2 ), B Upshot Recall that in calculus, you define the sum and scalar multiple offunctions pointwise, (f g )(x) f (x) g (x), (λf )(x) λf (x).The above formulas show that the linear function corresponding to A B whichsends x 7 (A B)x Ax Bx is the pointwise sum of the functionscorresponding to A and B.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Another distributive & associative lawPropositionFor A, B Mmn (R), λ R, x Rn(A B)x Ax Bx.(λA)x λ(Ax).Proof. Suppose n 2 (else need more space) so A (a1 a2 ), B Upshot Recall that in calculus, you define the sum and scalar multiple offunctions pointwise, (f g )(x) f (x) g (x), (λf )(x) λf (x).The above formulas show that the linear function corresponding to A B whichsends x 7 (A B)x Ax Bx is the pointwise sum of the functionscorresponding to A and B. The same goes for the scalar multiple.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20217 / 33
Basic properties of matrix arithmeticDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Basic properties of matrix arithmeticPropositionFor A, B Mmn , and scalars λ, µDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Basic properties of matrix arithmeticPropositionFor A, B Mmn , and scalars λ, µλ(µA) (λµ)A.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Basic properties of matrix arithmeticPropositionFor A, B Mmn , and scalars λ, µλ(µA) (λµ)A.(λ µ)A Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Basic properties of matrix arithmeticPropositionFor A, B Mmn , and scalars λ, µλ(µA) (λµ)A.(λ µ)A λ(A B) Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Basic properties of matrix arithmeticPropositionFor A, B Mmn , and scalars λ, µλ(µA) (λµ)A.(λ µ)A λ(A B) Proof. Just as for vectors e.g.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20218 / 33
Geometric example of linear combinations A 10 01 B 100 1 C : 12 (A B)Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 20219 / 33
Matrix multiplicationDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202110 / 33
Matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp .Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202110 / 33
Matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp . We define the matrix product AB to bethe m p-matrixAB (Ab1 . . . Abp ) Mmp .Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202110 / 33
Matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp . We define the matrix product AB to bethe m p-matrixAB (Ab1 . . . Abp ) Mmp . 0 1 1 1E.g. 2 3 1 24 5Alternatively, the ij-th entry of AB comes from “zipping up” the ith row of Awith the jth column of B: i.e. if A (aij ), B (bij ) b1jnX [AB]ij (ai1 . . . ain ) . ai1 b1j . . . ain bnj ail blj .l 1bnjDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202110 / 33
Matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp . We define the matrix product AB to bethe m p-matrixAB (Ab1 . . . Abp ) Mmp . 0 1 1 1E.g. 2 3 1 24 5Alternatively, the ij-th entry of AB comes from “zipping up” the ith row of Awith the jth column of B: i.e. if A (aij ), B (bij ) b1jnX [AB]ij (ai1 . . . ain ) . ai1 b1j . . . ain bnj ail blj .l 1bnjWarning The product AB is only defined when no. columns A no. rows B.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202110 / 33
Associative lawDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Associative lawAssociative law of matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp , C (c1 . . . cq ) Mpq .Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Associative lawAssociative law of matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp , C (c1 . . . cq ) Mpq . Then(AB)C A(BC ).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Associative lawAssociative law of matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp , C (c1 . . . cq ) Mpq . Then(AB)C A(BC ). c1 Proof. It suffices show this for C c . for assuming this case we seecpDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Associative lawAssociative law of matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp , C (c1 . . . cq ) Mpq . Then(AB)C A(BC ). c1 Proof. It suffices show this for C c . for assuming this case we seecp(AB)C ((AB)c1 . . . (AB)cq ) (A(Bc1 ) . . . A(Bcq )) A((Bc1 ) . . . (Bcq )) A(BC ).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Associative lawAssociative law of matrix multiplicationLet A Mmn , B (b1 . . . bp ) Mnp , C (c1 . . . cq ) Mpq . Then(AB)C A(BC ). c1 Proof. It suffices show this for C c . for assuming this case we seecp(AB)C ((AB)c1 . . . (AB)cq ) (A(Bc1 ) . . . A(Bcq )) A((Bc1 ) . . . (Bcq )) A(BC ).If C c then(AB)c (Ab1 . . . Abp )c c1 Ab1 . . . cp Abp A(c1 b1 . . . cp bp ) A(Bc).Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202111 / 33
Functional interpretation of the associative lawDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx)Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx) is the composite x 7 Bx 7 A(Bx) of the linear mapsassociated to A and B.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx) is the composite x 7 Bx 7 A(Bx) of the linear mapsassociated to A and B. 0E.g. Recall that B 10 1corresponds to reflection about the x-axis.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx) is the composite x 7 Bx 7 A(Bx) of the linear mapsassociated to A and B. 0E.g. Recall that B 10 1corresponds to reflection about the x-axis. Thefunctional viewpoint shows that B 2 coresponds to the mappingDaniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx) is the composite x 7 Bx 7 A(Bx) of the linear mapsassociated to A and B. 0E.g. Recall that B 10 1corresponds to reflection about the x-axis. Thefunctional viewpoint shows that B 2 coresponds to the mappingLet’s check1 00 1 Daniel Chan (UNSW)1 00 1 Chapter 5: MatricesTerm 1 202112 / 33
Functional interpretation of the associative lawThe associative law says the function associated to AB which mapsx 7 (AB)x A(Bx) is the composite x 7 Bx 7 A(Bx) of the linear mapsassociated to A and B. 0E.g. Recall that B 10 1corresponds to reflection about the x-axis. Thefunctional viewpoint shows that B 2 coresponds to the mappingLet’s check1 00 1 1 00 1 Remark The definition of matrix multiplication was designed so that it reflectsthe composition of linear maps.Daniel Chan (UNSW)Chapter 5: MatricesTerm 1 202112 / 33
Distributive laws & noncommutativityDaniel Chan (UNSW)Chapter 5: MatricesTerm
Matrices were rst introduced in the Chinese \Nine Chapters on the Mathematical Art" to solve linear eqns. In the mid-1800s, senior wrangler Arthur Cayley studied matrices in their own right and showed how they have an interesting and useful algebra associated to them. We will look at Cayley’s ideas and extend vector arithmetic to matrices and
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
22 Dense matrices over the Real Double Field using NumPy435 23 Dense matrices over GF(2) using the M4RI library437 24 Dense matrices over F 2 for 2 16 using the M4RIE library447 25 Dense matrices over Z/ Z for 223 using LinBox’s Modular double 455 26 Dense matrices over Z/ Z for 211 using LinBox’s Modular&l
Class XII – NCERT – Maths Chapter 3 - Matrices 3.Matrices . Exercise 3.1 . Question 1: In the matrix 2 5 19 7 5 35 2 12 2 3 1 5 17. A . As the given matrices are equal, their corresponding elements are also equal. Class XII – NCERT – Maths . Chapter 3 - Matrices . 3.Matrices . Comparing the corresponding elements, we get: .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
BLOSUM vs. PAM Equivalent PAM and BLOSUM matrices based on relative entropy PAM100 Blosum90 PAM120 Blosum80 PAM160 Blosum60 PAM200 Blosum52 PAM250 Blosum45 PAM matrices have lower expected scores for the BLOSUM matrices with the same entropy BLOSUM matrices “generally perform better” than PAM matrices
matrices with capital letters, like A, B, etc, although we will sometimes use lower case letters for one dimensional matrices (ie: 1 m or n 1 matrices). One dimensional matrices are often called vectors, as in row vector for a n 1 matrix or column vector for a 1 m matrix but we are going
Matrices and Determinants (i) Matrices Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a . Further Reduced ISC Class 12 Maths Syllabus 2020-21
