In our matrices class we were given a problem that I'm having trouble with.
Let $E$ and $H$ be subspaces of $\mathbb{C}^n$ such that $\mathbb{C}^n = E \oplus H$. Construct a projection $\mathbf P$ such that $R(\mathbf P) = E$ and $R(\mathbf I − \mathbf P) = H$. Hint: set $F = H ^\perp$ , take a basis $\mathbf v_1, \dots, \mathbf v_d$ of $E$, and use theorems 3.4 and 3.3.
Here are the related theorems:
So, apparently I need such $\mathbf P$ that $\mathbf P^2 = \mathbf P$. Also, it needs to have $R(\mathbf P) = E$ and $R(\mathbf I − \mathbf P) = H$. But I have no idea how to utilize the result we get from using theorem 3.4...
Any help would be appreciated.