Summary of MIT's Linear Algebra, Lecture 5: Vector Spaces and Subspaces

I wished it had included the proof that the inverse of a permutation matrix is its transpose. That fact is cute but is not entirely obvious. Since it didn't, I'll outline an easy proof for anyone who wishes to fill in the details.

1. The transpose of a permutation matrix is a permutation matrix.

2. The product of two permutation matrices is a permutation matrix.

3. If P is a permutation matrix and Q is its transpose, then from 1 we know that Q is a permutation matrix and from 2 we know that PQ is a permutation matrix.

4. The lecture showed that anything times its transpose is symmetric. So show that the only symmetric permutation matrix is the identity matrix and you've shown that PQ is the identity matrix. Therefore Q must be the inverse of P.

QED

wow that's so nice !! I'm actually studing that in my uni in France but my teacher is sooooo boring. Watching to those videos make me feel like I'm in his class. Real good occasion to profit from MIT's class without being at MIT ^^