Answer
False
Work Step by Step
Every $3 × 3$ upper triangular matrices are in form:
$\begin{bmatrix}
a & b & c \\ 0 & d & e\\ 0 & 0 & f
\end{bmatrix}$
We can see that set of all upper triangular matrices is 6-dimensional subspace of $M_3(R)$
Hence, the statement is false.
