有限维空间中商算子T/U的上三角矩阵证明问询（LADR4e 5C12(b)）

Hey everyone,

This is a problem from Linear Algebra Done Right (4th Edition), exercise 5C12 (b):

Suppose $V$ is finite-dimensional and $T \in L(V)$ has an upper-triangular matrix with respect to some basis of $V$. $U$ is a subspace of $V$ that is invariant under $T$.
Prove that the quotient operator $T/U$ has an upper-triangular matrix with respect to some basis of $V/U$.

I've been racking my brain over this problem for quite a while and came up with two possible approaches, but I've hit a wall with both:

• a) My first idea was to show that if we take a basis $w_1,w_2,\ldots,w_m$ of $U$ (with respect to which the restriction $T|_U$ has an upper-triangular matrix), we can extend this to a basis of $V$ while preserving some upper-triangular structure... (the rest of this thought got cut off)

备注：内容来源于stack exchange，提问作者Chenming Zhang