Exercise C16 Pg. 349

Find the Matrix representation of

\(T : C^3 \rightarrow C^4 , T\)

\[\begin{bmatrix} x\\ y\\ z \end{bmatrix}\] = \[\begin{bmatrix} 3x + 2y +z\\ x+y+z\\ x-3y\\ 2x+3y+z \end{bmatrix}\]

)

Using Theorom MLTCV - Matrix Linear Transformation, Column Vectors we can find the matrix representation

\(C_1 = T(e_1) =\)

[ \[\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}\]

=

\[\begin{bmatrix} 3*1 + 2*0 + 0\\ 1 + 0 + 0\\ 1 - (3 * 0)\\ (2*1) + (3 *0) + 0 \end{bmatrix}\] = \[\begin{bmatrix} 3\\ 1\\ 1\\ 2 \end{bmatrix}\]

]

\(C_2 = T(e_2) =\)

\[\begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}\]

=

\[\begin{bmatrix} (3*0) + (2*1) + 0\\ 0 + 1 + 0\\ 0 - (3 * 1)\\ (2*0) + (3*1) + 0 \end{bmatrix}\] = \[\begin{bmatrix} 2\\ 1\\ -3\\ 3 \end{bmatrix}\]

\(C_3 = T(e_3) =\)

\[\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\]

=

\[\begin{bmatrix} 3*0 + 2*0 + 1\\ 0 + 0 + 1\\ 0 - (3 * 0)\\ (2*0) + (3 *0) + 1 \end{bmatrix}\] = \[\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}\]

\(C_4 =\)

\[\begin{bmatrix} 3 & 2 & 1\\ 1 & 1 & 1\\ 1 & -3 & 0\\ 2 & 3 & 1 \end{bmatrix}\]

Check that \(T(x) = Cx\)

C <- matrix(c(3,2,1,1,1,1,1,-3,0,2,3,1), nrow = 4, byrow = T)

v <- matrix(c(-2, 1, 4), nrow = 3, byrow = T)

C %*% v
##      [,1]
## [1,]    0
## [2,]    3
## [3,]   -5
## [4,]    3
T_function <- function(x){
  m <- matrix(c(
    (3 * x[1,]) + (2 * x[2,]) +x[3,],
    x[1,] + x[2,] + x[3,],
    x[1,] - (3 * x[2,]), 
    (2 * x[1,]) + (3 * x[2,]) + x[3,]
  ))
}
print(T_function(v))
##      [,1]
## [1,]    0
## [2,]    3
## [3,]   -5
## [4,]    3
T_function(v) == C %*% v
##      [,1]
## [1,] TRUE
## [2,] TRUE
## [3,] TRUE
## [4,] TRUE