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

By Robert A. Beezer

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Then the conjugate of A, written A is an m × n matrix defined by A ij = [A]ij c 2005, 2006 Robert A. Beezer Theorem CRMA Conjugation Respects Matrix Addition 109 Suppose that A and B are m × n matrices. Then A + B = A + B. c 2005, 2006 Theorem CRMSM Robert A. Beezer Conjugation Respects Matrix Scalar Multiplication 110 Suppose that α ∈ C and A is an m × n matrix. Then αA = αA. c 2005, 2006 Robert A. Beezer Theorem CCM Conjugate of the Conjugate of a Matrix 111 Suppose that A is an m × n matrix.

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 , . . , αn−r ∈ C} c 2005, 2006 Robert A. Beezer Theorem PSPHS Particular Solution Plus Homogeneous Solutions 67 Suppose that w is one solution to the linear system of equations LS(A, b).

Fn−r } be the sets of column indices where B does and does not (respectively) 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 Define the set S = {z1 , z2 , z3 , . . , 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 , .

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