Chapter 1.2 MatrixAlgebra

wm where wi ai 1 ai 2 ain let x be a column

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Unformatted text preview: ing row vectors w1 , w2 , . . . , wm , where wi = [ ai 1 ai 2 · · · ain ]. Let x be a column vector of size n with components x1 , x2 , . . . , xn . The product of A and x is the column vector of size m given by a11 x1 + a12 x2 + · · · + a1n xn w1 x1 w2 x2 a21 x1 + a22 x2 + · · · + a2n xn Ax = . . = . . . . . . . wm xn am1 x1 + am2 x2 + · · · + amn xn Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose The Product as a Linear Combination of Columns Suppose A has column vectors v1 , v2 , . . . , vn , where a1i a2i vi = . . . ami We can write Ax as Ax = v1 v2 · · · vn x1 x2 . . . xn = x1 v1 + x2 v2 + · · · + xn vn . Outline Matrices Matrix Algebra Matrix Equations Examples Example 1 Compute Ax, where A= 2 2 −4 1 6 5 −10 Compute Ax, where 1 −1 3 0 1 A= 2 1 1 −2 3 and x = 2 −5 −1 and x = 5 2 Trace and Transpose Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Properties Theorem If A is an m × n matrix, x, y are column vectors of size n, and λ ∈ R is a scalar, then: 1 A(x + y) = Ax + Ay; 2 A(λx) = λAx....
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