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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:
Theorem 3.4 (and the related lemma) Theorem 3.3

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.

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1 Answer 1

If the basis is orthonormal, then consider $\sum_{j=1}^d |e_j\rangle \langle e_j|$ where $|e_j\rangle$ represents $e_j$ as a column vector and $\langle e_j|$ is its transpose.

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