veneciasp

2022-06-29

If the transformation is from ${\mathbb{R}}^{3}\to \mathbb{R}$ is
$T\left\{a,b,c\right\}={\int }_{0}^{\pi }2a{e}^{t}+2b\mathrm{sin}\left(t\right)+3c\mathrm{cos}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$
How to find the standard matrix?

Alexia Hart

Expert

By "standard matrix" maybe you mean the matrix with respect to the basis
$B=\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)\right\}$ for ${\mathbb{R}}^{3}$
We could rewrite $T$ this way:
$T:{\mathbb{R}}^{3}\to \mathbb{R}$
$T\left(v\right)=T\left(\left(x,y,z\right)\right)={\int }_{0}^{\pi }2x{e}^{t}+2ysin\left(t\right)+3zcos\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$
$=2x{\int }_{0}^{\pi }{e}^{t}\phantom{\rule{thinmathspace}{0ex}}dt+2y{\int }_{0}^{\pi }sin\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt+3z{\int }_{0}^{\pi }cos\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$
To write the matrix of a transformation with respect to some basis we should first apply the transformation on the basis vectors, so for example:
$T\left(\left(1,0,0\right)\right)=2\left(1\right){\int }_{0}^{\pi }{e}^{t}\phantom{\rule{thinmathspace}{0ex}}dt+2\left(0\right){\int }_{0}^{\pi }sin\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt+3\left(0\right){\int }_{0}^{\pi }cos\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$
$=2{\int }_{0}^{\pi }{e}^{t}\phantom{\rule{thinmathspace}{0ex}}dt$
$=2\left({e}^{\pi }-{e}^{0}\right)$
$=2\left({e}^{\pi }-1\right)$
To get the matrix you should apply $T$ to the other base vectors and form the matrix with 3 rows and 1 column.
$T\left(\left(0,1,0\right)\right)=2{\int }_{0}^{\pi }sin\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt=4$
$T\left(\left(0,0,1\right)\right)=3{\int }_{0}^{\pi }cos\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt=0$
So the matrix of $T$ with respect to $B$ is: $\left(\begin{array}{c}2\left({e}^{\pi }-1\right)\\ 4\\ 0\end{array}\right)$
To apply $T$ to any $\left(x,y,z\right)\in {\mathbb{R}}^{3}$ simply multiply:
$T\left(\left(x,y,z\right)\right)=\left(\begin{array}{ccc}x& y& z\end{array}\right)\left(\begin{array}{c}2\left({e}^{\pi }-1\right)\\ 4\\ 0\end{array}\right)$

gaiaecologicaq2

Expert