Emerson Barnes

2022-01-20

Find matrix of linear transformation
A linear transformation
$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$
is given by
$T\left(i\right)=i+j$
$T\left(j\right)=2i-j$

egowaffle26ic

Expert

Step 1
If and and
and , then
$T\left({e}_{1}\right)=T\left(i-j\right)$
$=T\left(i\right)-T\left(j\right)$

$={a}_{1}{e}_{1}+{b}_{1}{e}_{2}$ (say)
and
$T\left({e}_{2}\right)=T\left(3i+j\right)$

$={a}_{2}{e}_{1}+{b}_{2}{e}_{2}$ (say)
Thus the matrix of T w.r.t. the new basis is
$\left(\begin{array}{cc}{a}_{1}& {a}_{2}\\ {b}_{1}& {b}_{2}\end{array}\right)$
and you need to find the values of and ${b}_{2}$. The above systems of equations reduces to
${a}_{1}+2{b}_{1}=-1$
$-{a}_{1}+{b}_{1}=2$
and
${a}_{2}+3{b}_{2}=5$
$-{a}_{2}+{b}_{2}=2$
Solve these equations to obtain
${a}_{1}=-\frac{7}{4}$
${b}_{1}=\frac{1}{4}$
${a}_{2}=-\frac{1}{4}$
and
${b}_{2}=\frac{7}{4}$

suzzzlesv7

Expert

Step 1
The matrix of the transformation of basis is:
$P=\left[\begin{array}{cc}1& 3\\ -1& 1\end{array}\right]$
and
${P}^{-1}=\frac{1}{4}\left[\begin{array}{cc}1& -3\\ 1& 1\end{array}\right]$
${P}^{-1}TP=\frac{1}{4}\left[\begin{array}{cc}1& -3\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 1& -1\end{array}\right]\left[\begin{array}{cc}1& 3\\ -1& 1\end{array}\right]$
$=\frac{1}{4}\left[\begin{array}{cc}-7& -1\\ 1& 7\end{array}\right]$
Step 2
The figure illustrate how operate the given transformation in the two basis.
Indices refer to the canonical basis, indices to the new basis .
In the figure we have the vector $v={v}_{i,j}={\left[1,2\right]}^{T}$ (in canonical basis), that is transformed in ${v}^{\prime }={v}_{i,j}^{\prime }={\left[5,-1\right]}^{T}$
In the new basis we have:
${v}_{1,2}={P}^{-1}{v}_{i,j}=\frac{1}{4}\left[\begin{array}{c}-5\\ 3\end{array}\right]$
and, for the tranformed vector:
${v}_{1,2}^{\prime }={P}^{-1}{v}_{i,j}=\frac{1}{4}\left[\begin{array}{c}2\\ 1\end{array}\right]$
$={P}^{-1}{T}_{i.j}{v}_{i,j}$
$={P}^{-1}{T}_{i,j}P{v}_{1,2}$
$={T}_{1,2}{v}_{1,2}$
ao we have:
${T}_{1,2}={P}^{-1}{T}_{i,j}P=\frac{1}{4}\left[\begin{array}{cc}-7& -1\\ 1& 7\end{array}\right]$
I hope this can be helpful.

RizerMix

Expert

Step 1 ${e}_{1}+{e}_{2}=4i$ $i=\frac{{e}_{1}+{e}_{2}}{4}=P\left(i|j{\right)}^{T}$ ${e}_{2}-3{e}_{1}=4j$ $j=\frac{{e}_{2}-3{e}_{1}}{4}=P\left(i|j{\right)}^{T}$ Now compute the Matrix ${P}^{-1}$ and your T in the new Basis is given by ${P}^{-1}TP$