Answer to Question #265228 in Linear Algebra for Jannat Butt

Question #265228

Let

T:

R3→R2

be a linear transformation such that


(1,1,1) = (1,0),


(1,1,0) = (2, −1) and


(1,0,0) =


(4,3). What is


(2, −3,5)?


1
Expert's answer
2021-11-15T15:19:36-0500
T([111])=[10],T\bigg(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\bigg)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},

T([110])=[21],T\bigg(\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\bigg)=\begin{bmatrix} 2 \\ -1 \end{bmatrix},

T([100])=[63]T\bigg(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\bigg)=\begin{bmatrix} 6 \\ 3 \end{bmatrix}

A=[abcdef]A=\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}

a+b+c=1d+e+f=0\begin{matrix} a+b+c=1 \\ d+ e+f=0 \end{matrix}

a+b=2d+e=1\begin{matrix} a+b=2 \\ d+ e=-1 \end{matrix}

a=4d=3\begin{matrix} a=4 \\ d=3 \end{matrix}

A=[421341]A=\begin{bmatrix} 4 & -2 & -1 \\ 3 & -4 & 1 \end{bmatrix}

[421341][235]=[4(2)2(3)1(5)3(2)4(3)+1(5)]\begin{bmatrix} 4 & -2 & -1 \\ 3 & -4 & 1 \end{bmatrix}\begin{bmatrix} 2 \\ -3 \\ 5 \end{bmatrix}=\begin{bmatrix} 4(2)-2(-3)-1(5) \\ 3(2)-4(-3)+1(5) \end{bmatrix}

=[923]=\begin{bmatrix} 9 \\ 23 \end{bmatrix}


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