# Assume that T is a linear transformation. Find the standard

Assume that T is a linear transformation. Find the standard matrix of T.
T: , where ${e}_{1}=\left(1,0\right)$ and ${e}_{2}=\left(0,1\right)$
$A=$
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Joseph Fair
Let $x\in {F}^{n}$. Here, $\left[\begin{array}{c}{x}_{1}\\ ⋮\\ {x}_{n}\end{array}\right]$
$T\left(x\right)=T\left(\sum _{j}{e}_{j}{x}_{j}\right)=\sum _{j}T\left({e}_{j}\right){x}_{j}$
$=\sum _{j}{A}_{j}{x}_{j}=Ax$
So, here $A=\left[T\left({e}_{1}\right)\dots T\left({e}_{n}\right)\right]$
According to question
T:
So, the standart matrix of T is $\left[T\left({e}_{1}\right)T\left({e}_{2}\right)\right]$
$=\left[\begin{array}{cc}3& -5\\ 1& 6\\ 3& 0\\ 1& 0\end{array}\right]$ - Answer
###### Not exactly what you’re looking for?
Beverly Smith
A=$=\left[\begin{array}{cc}3& -5\\ 1& 6\\ 3& 0\\ 1& 0\end{array}\right]$
###### Not exactly what you’re looking for?
karton

let $T=\left[\begin{array}{cc}a& b\\ c& d\\ e& f\\ g& h\end{array}\right]$
$T\left({e}_{1}\right)=\left(3,1,3,1\right)$
$⇒T\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}3\\ 1\\ 3\\ 1\end{array}\right)$
$⇒\left[\begin{array}{cc}a& b\\ c& d\\ e& f\\ g& h\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 3\\ 1\end{array}\right]$
$⇒\left[\begin{array}{c}a\\ c\\ e\\ g\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 3\\ 1\end{array}\right]⇒a=3,c=1,e=3,g=1$
$T\left({e}_{2}\right)=\left(-5,6,0,0\right)$
$⇒T\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}-5\\ 6\\ 0\\ 0\end{array}\right)$
$⇒\left[\begin{array}{cc}a& b\\ c& d\\ e& f\\ g& h\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}-5\\ 6\\ 0\\ 0\end{array}\right]$
$⇒\left[\begin{array}{c}b\\ d\\ f\\ h\end{array}\right]==\left[\begin{array}{c}-5\\ 6\\ 0\\ 0\end{array}\right]⇒b=-5,d=6,f=h=0$
$⇒T=\left[\begin{array}{cc}3& -5\\ 1& 6\\ 3& 0\\ 1& 0\end{array}\right]$