A First Course in Linear Algebra - Flashcard Supplement by Robert A. Beezer

By Robert A. Beezer

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Extra resources for A First Course in Linear Algebra - Flashcard Supplement

Example text

B + C)D = BD + CD c 2005, 2006 Robert A. Beezer Theorem MMSMM Matrix Multiplication and Scalar Matrix Multiplication 125 Suppose A is an m × n matrix and B is an n × p matrix. Let α be a scalar. Then α(AB) = (αA)B = A(αB). c 2005, 2006 Theorem MMA Matrix Multiplication is Associative Robert A. Beezer 126 Suppose A is an m × n matrix, B is an n × p matrix and D is a p × s matrix. Then A(BD) = (AB)D. c 2005, 2006 Robert A.

Then x+y =x+y c 2005, 2006 Theorem CRSM Robert A. Beezer Conjugation Respects Vector Scalar Multiplication 82 Suppose x is a vector from Cm , and α ∈ C is a scalar. Then αx = α x c 2005, 2006 Robert A. Beezer Definition IP Inner Product 83 Given the vectors u, v ∈ Cm the inner product of u and v is the scalar quantity in C, m u, v = [u]1 [v]1 + [u]2 [v]2 + [u]3 [v]3 + · · · + [u]m [v]m = [u]i [v]i i=1 c 2005, 2006 Theorem IPVA Inner Product and Vector Addition Robert A. Beezer 84 Suppose u, v, w ∈ Cm .

Em } = { ej | 1 ≤ j ≤ m} is the set of standard unit vectors in Cm . c 2005, 2006 Robert A. Beezer Theorem OSLI Orthogonal Sets are Linearly Independent 93 Suppose that S is an orthogonal set of nonzero vectors. Then S is linearly independent. c 2005, 2006 Theorem GSP Robert A. Beezer Gram-Schmidt Procedure 94 Suppose that S = {v1 , v2 , v3 , . . , vp } is a linearly independent set of vectors in Cm . Define the vectors ui , 1 ≤ i ≤ p by ui = vi − vi , u2 vi , u3 vi , ui−1 vi , u1 u1 − u2 − u3 − · · · − ui−1 u1 , u1 u2 , u2 u3 , u3 ui−1 , ui−1 Then if T = {u1 , u2 , u3 , .

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