By Robert A. Beezer

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**Example text**

Beezer 102 Given an m × n matrix A, its transpose is the n × m matrix At given by At ij = [A]ji , 1 ≤ i ≤ n, 1 ≤ j ≤ m. c 2005, 2006 Robert A. Beezer Definition SYM Symmetric Matrix 103 The matrix A is symmetric if A = At . c 2005, 2006 Theorem SMS Symmetric Matrices are Square Robert A. Beezer 104 Suppose that A is a symmetric matrix. Then A is square. c 2005, 2006 Robert A. Beezer Theorem TMA Transpose and Matrix Addition 105 Suppose that A and B are m × n matrices. Then (A + B)t = At + B t .

Beezer More Vectors than Size implies Linear Dependence 74 Suppose that S = {u1 , u2 , u3 , . . , un } is the set of vectors in Cm , and that n > m. Then S is a linearly dependent set. 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.

Suppose that B has r nonzero rows, columns without leading 1’s with indices F = {f1 , f2 , f3 , . . , fn−r , n + 1}, and columns with leading 1’s (pivot columns) having indices D = {d1 , d2 , d3 , . . , dr }. Define vectors c, uj , 1 ≤ j ≤ n − r of size n by 0 if i ∈ F [B]k,n+1 if i ∈ D, i = dk if i ∈ F , i = fj 1 [uj ]i = 0 if i ∈ F , i = fj . − [B] if i ∈ D, i = dk k,fj [c]i = Then the set of solutions to the system of equations LS(A, b) is S = { c + α1 u1 + α2 u2 + α3 u3 + · · · + αn−r un−r | α1 , α2 , α3 , .