Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.