91影视

Consider the matrix of the given transformation be T

Let$T\left(\mathrm{f}\left(\mathrm{t}\right)\right)=f(\mathrm{t}-\raisebox{1ex}{$\mathrm{蟺}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)$be the transformation.

Assume$f\left(x\right)=a\cos \left(x\right)+b\sin \left(x\right)$be the function.

Hence$B=\left[{\left[T\left(\cos x\right)\right]}_B鈥�鈥�鈥�{\left[T\left(\sin x\right)\right]}_B鈥�\right]$

Then, the transformation be as follows.

$\begin{array}{rcl}T\cos \left(x\right)& =& \cos \left(x-\frac{蟺}{4}\right)\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \cos x\cos \frac{蟺}{4}+\sin x\sin \frac{蟺}{4}\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \cos x\frac{1}{\sqrt{2}}+\sin x\frac{1}{\sqrt{2}}\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x\\ & & \end{array}$

Similarly further simplification is as follows

$\begin{array}{rcl}T\left(\sin x\right)& =& \sin \left(x-\frac{蟺}{4}\right)\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \sin x\cos \frac{蟺}{4}-\cos x\sin \frac{蟺}{4}\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \sin x\frac{1}{\sqrt{2}}-\cos x\frac{1}{\sqrt{2}}\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \frac{1}{\sqrt{2}}\sin x-\frac{1}{\sqrt{2}}\cos x\\ & & \end{array}$

Therefore the matrix be B as follows

role="math" localid="1660420410409" $B=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$

Hence the solution.

Consider $B=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$be the matrix of the given transformation T.

$\begin{array}{rcl}\left[B\right]& =& \det \left(B\right)\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& ad-bc\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& \frac{1}{\sqrt{2}}脳\frac{1}{\sqrt{2}}-\left(-\frac{1}{\sqrt{2}}脳\frac{1}{\sqrt{2}}\right)\\ 鈥�鈥�鈥�鈥�鈥�鈥�鈥�& =& 1鈮�0\\ & & \end{array}$

Thus the transformation T is invertible as well.So T is an isomorphism.

Hence the solution.