Linear Algebra Linear transformation Help

Question

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear transformation, then there exists a basis for $\mathbb{R}^n$ in which $T$ is diagonal.

Is this true or false

Answer 1

No, assume $T$ represented by (in standard basis) $$\begin{pmatrix}1&1\\0&1\end{pmatrix}$$

There is no basis in which this matrix is diagonal.