By Robert A. Beezer

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

**Sample text**

Up . Symbolically, S = { α1 u1 + α2 u2 + α3 u3 + · · · + αp up | αi ∈ C, 1 ≤ i ≤ p} p αi ui αi ∈ C, 1 ≤ i ≤ p = i=1 c 2005, 2006 Robert A. Beezer Theorem SSNS Spanning Sets for Null Spaces 69 Suppose that A is an m × n matrix, and B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. Let D = {d1 , d2 , d3 , . . , dr } be the column indices where B has leading 1’s (pivot columns) and F = {f1 , f2 , f3 , . . , fn−r } be the set of column indices where B does not have leading 1’s.

2. S is a linearly independent set. c 2005, 2006 Theorem DLDS Dependency in Linearly Dependent Sets Robert A. 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.

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 , .

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

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