Assume that T is a linear transformation. Find the standard

Priscilla Johnston 2022-01-06 Answered
Assume that T is a linear transformation. Find the standard matrix of T. \(\displaystyle{T}:{R}^{{{2}}}\rightarrow{R}^{{{4}}},{T}{\left({e}_{{1}}\right)}={\left({3},{1},{3},{1}\right)}\) and \(\displaystyle{T}{\left({e}_{{2}}\right)}={\left(-{5},{2},{0},{0}\right)}\), where \(\displaystyle{e}_{{1}}={\left({1},{0}\right)}\) and \(\displaystyle{e}_{{2}}={90},{1}{)}\)

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Expert Answer

SlabydouluS62
Answered 2022-01-07 Author has 1786 answers
We know, that
\[e_1=\begin{bmatrix}1 \\0 \end{bmatrix}\]
\[e_2=\begin{bmatrix}0 \\1 \end{bmatrix}\]
The task is given
\[T(e_1)=\begin{bmatrix}3 \\1 \\3 \\1 \end{bmatrix}\]
\[T(e_2)=\begin{bmatrix}-5 \\2 \\0 \\0 \end{bmatrix}\]
There exists a unique matrix A for the linear transformation T for which it holds \(\displaystyle{T}{\left({u}\right)}={A}{u}\) for all u and A is a form \(\displaystyle{A}={\left[{T}{\left({e}_{{1}}\right)},\ldots,{T}{\left({e}_{{n}}\right)}\right]}\), where \(\displaystyle{e}_{{i}},{i}={1},{2},\ldots\) 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:
\[A=[T(e_1) T(e_2)]=\begin{bmatrix}3 & -5 \\1 & 2 \\3 & 0 \\1 & 0 \end{bmatrix}\]
\[A=\begin{bmatrix}3 & -5 \\1 & 2 \\3 & 0 \\1 & 0 \end{bmatrix}\]
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