By Robert A. Beezer

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C 2005, 2006 Robert A. Beezer Theorem NMLIC Nonsingular Matrices have Linearly Independent Columns 75 Suppose that A is a square matrix. Then A is nonsingular if and only if the columns of A form a linearly independent set. c 2005, 2006 Theorem NME2 Robert A. Beezer Nonsingular Matrix Equivalences, Round 2 76 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}.

Beezer Theorem AMSM Adjoint and Matrix Scalar Multiplication 115 ∗ Suppose α ∈ C is a scalar and A is a matrix. Then (αA) = αA∗ . c 2005, 2006 Theorem AA Adjoint of an Adjoint Robert A. Beezer 116 ∗ Suppose that A is a matrix. Then (A∗ ) = A c 2005, 2006 Robert A. Beezer Definition MVP Matrix-Vector Product 117 Suppose A is an m × n matrix with columns A1 , A2 , A3 , . . , An and u is a vector of size n. Then the matrix-vector product of A with u is the linear combination Au = [u]1 A1 + [u]2 A2 + [u]3 A3 + · · · + [u]n An c 2005, 2006 Theorem SLEMM Robert A.

Then the matrix-vector product of A with u is the linear combination Au = [u]1 A1 + [u]2 A2 + [u]3 A3 + · · · + [u]n An c 2005, 2006 Theorem SLEMM Robert A. Beezer Systems of Linear Equations as Matrix Multiplication 118 The set of solutions to the linear system LS(A, b) equals the set of solutions for x in the vector equation Ax = b. c 2005, 2006 Robert A. Beezer Theorem EMMVP Equal Matrices and Matrix-Vector Products 119 Suppose that A and B are m × n matrices such that Ax = Bx for every x ∈ Cn .