Lipschitz Continuity of Linear Map Between Finite Dimensional Vector Spaces

There are a few ways you could proceed. For example, it is equivalent to showing that the following quantity is bounded: $\sup_{z\in X, \|z\| = 1}\|Tz\|$. Indeed, $\|T(x) - T(y)\|\leq \lambda\|x - y\|$ if and only if

$$ \frac{\|T(x) - T(y)\}}{\|x - y\|}\leq \lambda, $$

assuming that $x\neq y$. Now by linearity of $T$, the above inequality holds if and only if

$$ \left\|T\left(\frac{x - y}{\|x - y\|}\right)\right\|\leq \lambda. $$

Rewriting this with $z = x - y$, we get the claim.

To bound this, notice that $\|Tz\|\leq (|a_1| + \cdots + |a_n|)\max_{1\leq i\leq n}\|Te_i\|$, where $\{e_i\}$ is an orthonormal basis for $X$ and $z = \sum_{i=1}^n a_i e_i$. Since $e_1,\dots,e_n$ are orthonormal and $\|z\| = 1$, this puts a uniform (i.e. independent of $z$) bound on $|a_1|+\cdots+|a_n|$ (prove this).