By Robert A. Beezer

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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. Construct the n − r vectors zj , 1 ≤ j ≤ n − r of size n as if i ∈ F , i = fj 1 [zj ]i = 0 if i ∈ F , i = fj − [B] if i ∈ D, i = dk k,fj Then the null space of A is given by N (A) = {z1 , z2 , z3 , . . , zn−r } c 2005, 2006 Definition RLDCV Robert A. Beezer Relation of Linear Dependence for Column Vectors 70 Given a set of vectors S = {u1 , u2 , u3 , .

Then the conjugate of the vector, u, is defined by [u]i = [u]i 1≤i≤m c 2005, 2006 Robert A. Beezer Theorem CRVA Conjugation Respects Vector Addition 81 Suppose x and y are two vectors from Cm . Then x+y =x+y c 2005, 2006 Theorem CRSM Robert A. Beezer Conjugation Respects Vector Scalar Multiplication 82 Suppose x is a vector from Cm , and α ∈ C is a scalar. Then αx = α x c 2005, 2006 Robert A. Beezer Definition IP Inner Product 83 Given the vectors u, v ∈ Cm the inner product of u and v is the scalar quantity in C, m u, v = [u]1 [v]1 + [u]2 [v]2 + [u]3 [v]3 + · · · + [u]m [v]m = [u]i [v]i i=1 c 2005, 2006 Theorem IPVA Inner Product and Vector Addition Robert A.

Beezer Linearly Independent Vectors and Homogeneous Systems 72 Suppose that A is an m × n matrix and S = {A1 , A2 , A3 , . . , An } is the set of vectors in Cm that are the columns of A. Then S is a linearly independent set if and only if the homogeneous system LS(A, 0) has a unique solution. c 2005, 2006 Robert A. Beezer Theorem LIVRN Linearly Independent Vectors, r and n 73 Suppose that A is an m × n matrix and S = {A1 , A2 , A3 , . . , An } is the set of vectors in Cm that are the columns of A.

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

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