Priscilla Johnston

2022-01-06

Assume that T is a linear transformation. Find the standard matrix of T. $T:{R}^{2}\to {R}^{4},T\left({e}_{1}\right)=\left(3,1,3,1\right)$ and $T\left({e}_{2}\right)=\left(-5,2,0,0\right)$, where ${e}_{1}=\left(1,0\right)$ and ${e}_{2}=90,1\right)$

SlabydouluS62

We know, that
${e}_{1}=\left[\begin{array}{c}1\\ 0\end{array}\right]$
${e}_{2}=\left[\begin{array}{c}0\\ 1\end{array}\right]$
$T\left({e}_{1}\right)=\left[\begin{array}{c}3\\ 1\\ 3\\ 1\end{array}\right]$
$T\left({e}_{2}\right)=\left[\begin{array}{c}-5\\ 2\\ 0\\ 0\end{array}\right]$
There exists a unique matrix A for the linear transformation T for which it holds $T\left(u\right)=Au$ for all u and A is a form $A=\left[T\left({e}_{1}\right),\dots ,T\left({e}_{n}\right)\right]$, where ${e}_{i},i=1,2,\dots$ are vectors from the identity matrix, respectively to columns.
Matrix A is the standart matrix for the linear transformation T.
So, the standart form is:
$A=\left[T\left({e}_{1}\right)T\left({e}_{2}\right)\right]=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$
$A=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$

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