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

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By Robert A. Beezer

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N (A) = {0}. c 2005, 2006 Robert A. Beezer Theorem NMUS Nonsingular Matrices and Unique Solutions 57 Suppose that A is a square matrix. A is a nonsingular matrix if and only if the system LS(A, b) has a unique solution for every choice of the constant vector b. c 2005, 2006 Theorem NME1 Robert A. Beezer Nonsingular Matrix Equivalences, Round 1 58 Suppose that A is a square matrix. The following are equivalent. 1. A is nonsingular. 2. A row-reduces to the identity matrix. 3. The null space of A contains only the zero vector, N (A) = {0}.

AIM Additive Inverses, Matrices If A ∈ Mmn , then there exists a matrix −A ∈ Mmn so that A + (−A) = O. • SMAM Scalar Multiplication Associativity, Matrices If α, β ∈ C and A ∈ Mmn , then α(βA) = (αβ)A. • DMAM Distributivity across Matrix Addition, Matrices If α ∈ C and A, B ∈ Mmn , then α(A + B) = αA + αB. • DSAM Distributivity across Scalar Addition, Matrices If α, β ∈ C and A ∈ Mmn , then (α + β)A = αA + βA. • OM One, Matrices If A ∈ Mmn , then 1A = A. c 2005, 2006 Robert A. Beezer Definition ZM Zero Matrix 101 The m × n zero matrix is written as O = Om×n and defined by [O]ij = 0, for all 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Beezer 78 Suppose that S = {u1 , u2 , u3 , . . , un } is a set of vectors. Then S is a linearly dependent set if and only if there is an index t, 1 ≤ t ≤ n such that ut is a linear combination of the vectors u1 , u2 , u3 , . . , ut−1 , ut+1 , . . , un . c 2005, 2006 Robert A. Beezer Theorem BS Basis of a Span 79 Suppose that S = {v1 , v2 , v3 , . . , vn } is a set of column vectors. Define W = S and let A be the matrix whose columns are the vectors from S. Let B be the reduced row-echelon form of A, with D = {d1 , d2 , d3 , .

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