Chapter 5.5, Problem 1E

1. Let $V$ be the set of all real $\left(3\times 3\right)$ matrices, and let $V_1\text{ and }{V}_2$ be subsets of V,

where $V_1$ consists of all the $\left(3\times 3\right)$ lower-triangular matrices and $V_2$ consists of all

the $\left(3\times 3\right)$ upper-triangular matrices.

a) Show that $V_1\text{ and }{V}_2$ are subspaces of $V$.

b) Find bases for $V_1\text{ and }{V}_2$.

c) Calculate $\dim \left(V\right),\dim \left(V_1\right)\text{ and dim}\left(V_2\right)$