Linear Algebra Notes 1 - Linear Algebra Notes from Kolman...

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Linear Algebra Notes from Kolman and Hill, 9th edition: An mxn linear system Ax = b is a system of m linear equations in n unknowns x i , i=1,...,n. If the linear system has a solution it is consistent, otherwise it is inconsistent . Ax = 0 is a homogeneous linear system. The homogeneous linear system always has the trivial solution x = 0. The linear systems Ax = b and Cx = d are equivalent , if they both have exactly the same solutions. Def 1.1: An mxn matrix A is a rectangular array of mn real or complex numbers arranged in m horizontal rows and n vertical columns. Def 1.2: Two mxn matrices A=[a ij ] and B=[b ij ] are equal , if they agree entry by entry. Def 1.3: The mxn matrices A and B are added entry by entry. Def 1.4: If A=[a ij ] and r is a real number then the scalar multiple of A is the matrix rA=[ra ij ]. If A 1 , A 2 , ..., A k are mxn matrices and c 1 , c 2 , ..., c k are real numbers then an expression of the form c 1 A 1 + c 2 A 2 + ... + c k A k is a linear combination of the A's with coefficients c 1 , c 2 , ..., c k . Def 1.5: The transpose of the mxn matrix A=[a ij ] is the nxm matrix A T =[a ji ]. Def 1.6: The dot product or inner product of the n-vectors a=[a i ] and b=[b i ] is ab = a 1 b 1 + a 2 b 2 + ... + a n b n . Def 1.7: If A=[a ij ] is an mxp matrix and B=[b ij ] a pxn matrix they can be multiplied and the ij entry of the mxn C = AB: c ij = dot product of the (ith row of A) T with (jth column of B). Note: Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A with coefficients x i . Theorem 1.1 Let A, B, and C be mxn matrices, then (a) A + B = B + A (b) A + (B + C) = (A + B) + C (c) there is a unique mxn matrix O such that for any mxn matrix A A + O = A (d) for each mxn matrix A, there is aunique mxn matrix D such that A + D = O D = -A is the negative of A. Theorem 1.2 Let A, B, and C be matrices of the appropriate sizes, then (a) A(BC) = (AB)C (b) (A + B)C = AC + BC (c) C(A + B) = CA + CB Theorem 1.3 Let r, s be real numbers and A, B matrices of the appropriate sizes, then (a) r(sA) = (rs)A
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(b) (r + s)A = rA + sA (c) r(A + B) = rA + rB (d) A(rB) = r(AB) = (rA)B Theorem 1.4 Let r be a scalar, A, B matrices of appropriate sizes, then (a) (A T ) T =A (b) (A + B) T = A T + B T (c) (AB) T = B T A T (d) (rA) T = rA T Note: (a) AB need not equal BA (b) AB may be the zero matrix with A not equal O and B not equal O (c) AB may equal AC with B not equal C Def 1.8: A matrix A with real enties is symmetric if A T = A.
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