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MATA23H3 (77)
Lecture 3

# week 3 Lecture notes.pdf

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School
University of Toronto Scarborough
Department
Mathematics
Course
MATA23H3
Professor
Kathleen Smith
Semester
Winter

Description
Deﬁnition: Let A = a i‘ ∈ M n,kR) and B = b ‘j ∈ M k,m(R). The matrix product C = AB is the n × m matrix C = c ij where k X cij ai‘ ‘ja bi1 1jb + i2 2j a b ik kj= a ib j ‘=1 th th (a is the i row of A and b ij the j column of B.) Deﬁnition: The n×n identity matrix, denoted I (or I ) is deﬁned n by I = δ . ij Deﬁnition:IfA = a ij ∈ M n,nR), the elementsaii i = 1,2,··· ,n are called the diagonal elements or the diagonal of A. The trace of A, denoted tr(A), is n X tr(A) = a iia 11 + a22 + ··· + a .nn i=1 A square matrix with all zero elements, except possibly on the diag- onal is called a diagonal matrix. Deﬁnition: If A = aij ∈ M n,kR), then the matrix aj i ∈ M (R) is called the transpose of A and is denoted A . k,n T Deﬁnition: Let A ∈ M n,nR). If A = A , then A is said to be T symmetric. If A = −A , then A is said to be skew-symmetric. Properties of matrix multiplication (when deﬁned) A(B C) = (AB)C I A = A B I = B A(B + C) = AB + AC (A + B)C = AC + B C λ(AB) = (λA)B = A(λB) Properties of transpose (when deﬁned) T T (A ) = A (A + B) = A + B T (AB) = B A T T Properties of trace (when deﬁned) tr(A + B) = tr(A) + tr(B) tr(λA) = λ tr(A) tr(AT) = tr(A) tr(AB) = tr(B A) Given the system of linear equations a x +a x +··· +a x = b 11 1 12 2 1k k 1 a21 1+a 22 2·· +a x 2k k= b 2 . . . an1 1a xn2 2· +a x =nk k n Think of the coeﬃcienti of x as being a column vector and the sys- tem as linear combinations of these column vectors. We can write as the matrix equation Ax = b,        a11 a12··· a1k  x1  b1              a21 a22··· a2k  x2  b2 where A =  , x =  and b = .  .   .   .   .   .   .        a a ··· a x b n1 n2 nk k n A is called the coeﬃcient matrix of the system. k Any v ∈ R such that Av = b is called a solution of the system. To solve the system, we combine A and b into a matrix,    a11 a12 ··· a1k b1    
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