Read e-book online A First Course in Linear Algebra - Flashcard Supplement PDF

By Robert A. Beezer

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Un from Cm and n scalars α1 , α2 , α3 , . . , αn , their linear combination is the vector α1 u1 + α2 u2 + α3 u3 + · · · + αn un c 2005, 2006 Robert A. Beezer Theorem SLSLC Solutions to Linear Systems are Linear Combinations 65 Denote the columns of the m × n matrix A as the vectors A1 , A2 , A3 , . . , 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 Definition NV Norm of a Vector 87 The norm of the vector u is the scalar quantity in C m u = 2 2 2 2 2 |[u]1 | + |[u]2 | + |[u]3 | + · · · + |[u]m | = |[u]i | i=1 c 2005, 2006 Theorem IPN Inner Products and Norms Suppose that u is a vector in Cm . Then u 2 Robert A. Beezer 88 = u, u . c 2005, 2006 Robert A. Beezer Theorem PIP Positive Inner Products 89 Suppose that u is a vector in Cm . Then u, u ≥ 0 with equality if and only if u = 0. c 2005, 2006 Definition OV Orthogonal Vectors Robert A.

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. Then A = A. c 2005, 2006 Theorem MCT Matrix Conjugation and Transposes Robert A. Beezer 112 t Suppose that A is an m × n matrix. Then (At ) = A . c 2005, 2006 Robert A. Beezer Definition A Adjoint 113 t If A is a matrix, then its adjoint is A∗ = A .

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


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