Aron Campbell

2023-03-30

T must be a linear transformation, we assume. Can u find the T standard matrix.$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$

?Donna Schmidt

Beginner2023-03-31Added 4 answers

To find the standard matrix for the linear transformation T: ${\mathbb{R}}^{2}\to {\mathbb{R}}^{4}$, we need to determine how T maps the standard basis vectors ${\mathbf{e}}_{1}$ and ${\mathbf{e}}_{2}$ of ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{4}$.

Given:

$T({\mathbf{e}}_{1})=(3,1,3,1)$

$T({\mathbf{e}}_{2})=(-5,2,0,0)$

The standard matrix for T can be formed by arranging the images of the basis vectors as columns.

$A=[T({\mathbf{e}}_{1})\phantom{\rule{0.167em}{0ex}}T({\mathbf{e}}_{2})]$

Substituting the given values, we have:

$A=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$

Therefore, the standard matrix for the linear transformation T is:

$A=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$

Given:

$T({\mathbf{e}}_{1})=(3,1,3,1)$

$T({\mathbf{e}}_{2})=(-5,2,0,0)$

The standard matrix for T can be formed by arranging the images of the basis vectors as columns.

$A=[T({\mathbf{e}}_{1})\phantom{\rule{0.167em}{0ex}}T({\mathbf{e}}_{2})]$

Substituting the given values, we have:

$A=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$

Therefore, the standard matrix for the linear transformation T is:

$A=\left[\begin{array}{cc}3& -5\\ 1& 2\\ 3& 0\\ 1& 0\end{array}\right]$

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