Chapter 1.2 MatrixAlgebra

# xm and b b1 b2 bn then ax

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Unformatted text preview: Matrix Equations Set x= x1 x2 . . . xm and b = b1 b2 . . . bn Then Ax = x1 a11 + x2 a12 + · · · + xm a1m x1 a21 + x2 a22 + · · · + xm a2m . . . .. . . . . . . . x1 an1 + x2 an2 + · · · + xm anm = Thus, the linear system of equations we started with is equivalent to the matrix equation Ax = b. b1 b2 . . . bn =b Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Matrix Equations If v1 , v2 , . . . , vm are the column vectors of A, then Ax = b has the form x1 v1 + x2 v2 + · · · + xm vm = b. Thus, ﬁnding a solution to the linear system of equations is equivalent to ﬁnding a solution to the matrix equation Ax = b. This, in turn, is equivalent to expressing b as a linear combination of the column vectors of A. Outline Matrices Matrix Algebra Matrix Equations Examples Example Express the system x1 − x2 + 3x3 − 2x4 = 8 5x1 + x3 + 4x4 = −1 −x1 + 2x2 − 7x3 + x4 = 3 a...
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