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C 2005, 2006 Robert A. Beezer Theorem EMP Entries of Matrix Products 121 Suppose A is an m × n matrix and B is an n × p matrix. Then for 1 ≤ i ≤ m, 1 ≤ j ≤ p, the individual entries of AB are given by [AB]ij = [A]i1 [B]1j + [A]i2 [B]2j + [A]i3 [B]3j + · · · + [A]in [B]nj n = [A]ik [B]kj k=1 c 2005, 2006 Theorem MMZM Matrix Multiplication and the Zero Matrix Robert A. Beezer 122 Suppose A is an m × n matrix. Then 1. AOn×p = Om×p 2. Op×m A = Op×n c 2005, 2006 Robert A. Beezer Theorem MMIM Matrix Multiplication and Identity Matrix 123 Suppose A is an m × n matrix.

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 , . . , dr } the set of column indices corresponding to the pivot columns of B. Then 1. T = {vd1 , vd2 , vd3 , . . vdr } is a linearly independent set. 2. W = T . c 2005, 2006 Definition CCCV Robert A.

Un } is a set of vectors from Cm . Then S is an orthogonal set if every pair of different vectors from S is orthogonal, that is ui , uj = 0 whenever i = j. c 2005, 2006 Definition SUV Standard Unit Vectors Robert A. Beezer 92 Let ej ∈ Cm , 1 ≤ j ≤ m denote the column vectors defined by [ej ]i = 0 if i = j 1 if i = j Then the set {e1 , e2 , e3 , . . , 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.

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

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