York

2020-11-30

Consider $V=\mathrm{cos}\left(x\right),\mathrm{sin}\left(x\right)$ a subspace of the vector space of continuous functions and a linear transformation $T:V\to V$ where $T\left(f\right)=f\left(0\right)×\mathrm{cos}\left(x\right)-f\left(\pi 2\right)×\mathrm{sin}\left(x\right).$

Find the matrix of T with respect to the basis $\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right),\mathrm{cos}\left(x\right)-\mathrm{sin}\left(x\right)$ and determine if T is an isomorphism.

Aamina Herring

The linear transformation$T\left(f\right)=f\left(0\right)\mathrm{cos}x-f\left(\pi \right)\mathrm{sin}x.$
$B=\mathrm{cos}x+\mathrm{sin}x,\mathrm{cos}x-\mathrm{sin}x.$
When $f\left(x\right)=\mathrm{cos}x+\mathrm{sin}xf\left(x\right)=\mathrm{cos}x+\mathrm{sin}x$,

$T\left(\mathrm{cos}x+\mathrm{sin}x\right)=\mathrm{cos}x+\mathrm{cos}x=2\mathrm{cos}x=\left(\mathrm{cos}x+\mathrm{sin}x\right)+\left(\mathrm{cos}x-\mathrm{sin}x\right).$
When $f\left(x\right)=\mathrm{cos}x-\mathrm{sin}xf\left(x\right)=\mathrm{cos}x-\mathrm{sin}x$,
$T\left(\mathrm{cos}x-\mathrm{sin}x\right)=\mathrm{cos}x+\mathrm{cos}x=2\mathrm{cos}x=\left(\mathrm{cos}x+\mathrm{sin}x\right)+\left(\mathrm{cos}x-\mathrm{sin}x\right).$
The matrix representation of T $\left[T\right]b=\left[1,1,1,1\right]$
Since $det\left(\left[T\right]B\right)\right)=0$, thus T is not an isomorphism.

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