Exercise LT.C20

Let

\[W = \begin{bmatrix} -3 \\ 1 \\ 4 \\ \end{bmatrix} \]

Suppose
S:C3→C4 is defined by

\[S =( \begin{bmatrix} X1 \\ X2 \\ X3 \\ \end{bmatrix} ) = \begin{bmatrix} 3X1 - 2X2 +5X3 \\ X1 + X2 + X3\\ 9X1 - 2X2 + 5X3 \\ 0X1 + 4X2 + 0X3 \\ \end{bmatrix} \]

compute S(w) two different ways.

1

w <- matrix(c(-3,1,4), nrow = 3)
w
##      [,1]
## [1,]   -3
## [2,]    1
## [3,]    4
s <- matrix(c(3,1,9,0,-2,1,-2,4,5,1,5,0), nrow = 4)
s
##      [,1] [,2] [,3]
## [1,]    3   -2    5
## [2,]    1    1    1
## [3,]    9   -2    5
## [4,]    0    4    0
s%*%w
##      [,1]
## [1,]    9
## [2,]    2
## [3,]   -9
## [4,]    4

2

\[S = \begin{bmatrix} X1 \\ X2 \\ X3 \\ \end{bmatrix} = \begin{bmatrix} -3 \\ 1 \\ 4 \\ \end{bmatrix} \]

\(3X1 - 2X2 + 5X3\)

= \(3*-3 -2*1 +5*4\)

= 9

\(X1 + X2 + X3\)

= \(1*-3 +1*1 + 1*4\)

= 2

\(9X1 - 2X2 + 5X3\)

= \(9*-3 -2*1 +5*4\)

= -9

\(0X1 + 4X2 + 0X3\)

= \(4*1\)

= 4