By Robert A. Beezer

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

**Example text**

Zn−r }. Then 1. N (A) = 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.

An . Then x is a solution to the linear system of equations LS(A, b) if and only if b equals the linear combination of the columns of A formed with the entries of x, [x]1 A1 + [x]2 A2 + [x]3 A3 + · · · + [x]n An = b c 2005, 2006 Theorem VFSLS Robert A. Beezer Vector Form of Solutions to Linear Systems 66 Suppose that [ A | b] is the augmented matrix for a consistent linear system LS(A, b) of m equations in n variables. Let B be a row-equivalent m × (n + 1) matrix in reduced rowechelon form.

Beezer Vector Form of Solutions to Linear Systems 66 Suppose that [ A | b] is the augmented matrix for a consistent linear system LS(A, b) of m equations in n variables. Let B be a row-equivalent m × (n + 1) matrix in reduced rowechelon form. 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 .

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

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