Prove/disprove that we cannot have $V = U_1 \oplus U_2$

Question

$V$ vector space of upper triangular $3\times3$ matrices.

$1$ and $2$ are subspaces of $$.

Every non-zero member of $1$ is invertible. Every member of $2$ is non-invertible.

Prove/disprove that we cannot have $=1⊕2$

I think that this is correct and

My idea is to prove that $U1 \cap U2 \ne {\{0}\}$

But I"m having hard time to write General member for $U1$ and $U2$

any hint how to prove this ?

thanks

Answer 1

Let $U_1=\operatorname{span}(I_3)$ and $U_2$ be the set of all upper triangular $3\times3$ matrices whose bottom rows are zero. Then $V=U_1\oplus U_2$.