Week 4 Discussion: Pg.444 Linear Transformations and Representations C41
C41. If \(T:C^2 \rightarrow C^3\) satisfies T(\(\begin{bmatrix}2\\3\end{bmatrix}) = \begin{bmatrix}2\\2\\1\end{bmatrix}\) and T(\(\begin{bmatrix}3\\4\end{bmatrix}) = \begin{bmatrix}-1\\0\\2\end{bmatrix}\), find the matrix representation of \(T\).
- find \(x_1\), \(x_2\) where \(\begin{bmatrix}2 & 3\\3 & 4\end{bmatrix}\times\begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix} = e_1\)
and find \(x_3\), \(x_4\) where \(\begin{bmatrix}2 & 3\\3 & 4\end{bmatrix}\times\begin{bmatrix}x_3\\x_4\end{bmatrix} = \begin{bmatrix}0\\1\end{bmatrix} = e_2\)
library(pracma)
mtrix <- matrix(c(2,3,3,4),byrow = TRUE,2,2)
c1 <- c(1,0)
c2 <- c(0,1)
rref(cbind(mtrix,c1))
## c1
## [1,] 1 0 -4
## [2,] 0 1 3
rref(cbind(mtrix,c2))
## c2
## [1,] 1 0 3
## [2,] 0 1 -2
\(\begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}-4\\3\end{bmatrix}\) and \(\begin{bmatrix}x_3\\x_4\end{bmatrix} = \begin{bmatrix}3\\-2\end{bmatrix}\)
- find \(T(\begin{bmatrix}x_1\\x_2\end{bmatrix}) = T(-4\begin{bmatrix}2\\3\end{bmatrix}+3\begin{bmatrix}3\\4\end{bmatrix}) = -4T(\begin{bmatrix}2\\3\end{bmatrix}) + 3T(\begin{bmatrix}3\\4\end{bmatrix}) = -4\begin{bmatrix}2\\2\\1\end{bmatrix} + 3\begin{bmatrix}-1\\0\\2\end{bmatrix}\)
and \(T(\begin{bmatrix}x_3\\x_4\end{bmatrix}) = T(3\begin{bmatrix}2\\3\end{bmatrix}-2\begin{bmatrix}3\\4\end{bmatrix}) = 3T(\begin{bmatrix}2\\3\end{bmatrix}) -2T(\begin{bmatrix}3\\4\end{bmatrix}) = 3\begin{bmatrix}2\\2\\1\end{bmatrix} - 2\begin{bmatrix}-1\\0\\2\end{bmatrix}\)
v1 <- c(-4,3)
v2 <- c(3,-2)
mtrix2 <- matrix(c(2,2,1,-1,0,2),byrow = FALSE,3,2)
reps1 <- mtrix2%*%v1
reps2 <- mtrix2%*%v2
- combine both vectors to get the matrix representation of \(T\)
answer <- cbind(reps1,reps2)
- verify the answer
answer%*%mtrix
## [,1] [,2]
## [1,] 2 -1
## [2,] 2 0
## [3,] 1 2