7 projection on a different axes question

[nhrock3]

7)
$$T:R^{2}->R^{2}$$ projection transformation on X-axes parallel to the
line
$$y=-\sqrt{3}x$$
find the representative matrices of T{*} by B=\{(1,0),(0,1)\} basis
how i tried:
i understood that the x axes stayed the same but the y axes turned
into
$$y=-\sqrt{3}x$$
our T takes some vector and returns only the new x part with respect
to the new y axes.
any guidanse?

 

[Mark44]

7)
$$T:R^{2}->R^{2}$$ projection transformation on X-axes parallel to the
line
$$y=-\sqrt{3}x$$
find the representative matrices of T{*} by B=\{(1,0),(0,1)\} basis
how i tried:
i understood that the x axes stayed the same but the y axes turned
into
$$y=-\sqrt{3}x$$
our T takes some vector and returns only the new x part with respect
to the new y axes.
any guidanse?

If I understand what you're trying to say, then
$$T\begin{bmatrix}1 \\ 0\end{bmatrix} = \begin{bmatrix}1\\ 0\end{bmatrix}$$
and
$$T\begin{bmatrix}0 \\ 1\end{bmatrix} = k\begin{bmatrix}1\\\sqrt{3}\end{bmatrix}$$

With a little trig you can figure out what k needs to be. What you know what a linear transformation does to a basis, you can write the matrix that represents the transformation.