# Basis of a transformation matrix for diagonal matrix

So I have this question here which says:

Let $$T:\Bbb R^3\to\Bbb R^3$$ defined by

$$T\left(\begin{matrix}x_1 \\ x_2 \\ x_3 \end{matrix}\right)=\left(\begin{matrix}-x_1+7x_2-x_3 \\ x_2 \\ 15x_2-2x_3 \end{matrix}\right).$$

• (a) Let $$\mathcal E=\{e_1,e_2,e_3\}$$, where $$e1=\left(\begin{matrix}1 \\ 0 \\ 0 \end{matrix}\right), e2=\left(\begin{matrix}0 \\ 1 \\ 0 \end{matrix}\right), e3=\left(\begin{matrix}0 \\ 0 \\ 1 \end{matrix}\right)$$, be the standard basis of $$\Bbb R^3$$. Find $$[T]_\mathcal E$$.
• (b) Find a basis $$B=\{b_1,b_2,b_3\}$$ of $$\Bbb R^3$$ such that $$[T]_B$$ is a diagonal matrix.
• (c) What is $$[T]_B$$?

I already solved part $$(a)$$. The answer is:

$$[T]_{\mathcal E} =\left( \begin{array}{ccc} -1 & 7 & -1\\ 0 & 1 & 0 \\ 0 & 15 & -2\\ \end{array} \right)$$

However, I'm not sure what the difference between part $$(b)$$ and part $$(c)$$ is. If I find a basis, isn't that automatically giving me $$[T]_{B}$$?

Second question. How do I actually do part $$(b)$$? As far as I can tell, I just have to diagonalize $$[T]_{B}$$ by finding the eigenvalues and the eigenvectors since that will give me a basis for the eigenspace which in turn, is a diagonal matrix. Is that all though? Is there more that I have to do?

• You are right on track. – xbh Dec 11 '18 at 6:55

Yes, all you have to do do solve part (b) is to find eigenvalues and eigenvectors for $$T$$. And then the answer to (c) is the diagonal matrix such tha the entries of the main diagonal are the eigenvalues that you got while solving (b).