So, I did pretty well for this course. There were six questions, summarised as follows (obviously I don’t remember the details):
1. Prove that multiplication by elementary matrices is equivalent to row operations in Gaussian elimination.
2. For a real inner product space with inner product (.,.) and induced norm |.|, prove the identity 2(|x|^2 + |y|^2) = |x-y|^2 + |x+y|^2
3. Find the normalised and unnormalised QR decomposition of a matrix.
4. a) Find the change of basis matrix and its inverse for a given vector space and two bases.
b) Prove one norm is equal to another norm, given two norms.
c) Calculate the norm of some linear map.
5. Prove that if p(x) is a polynomial and h is an eigenvalue of the matrix A, then p(h) is an eigenvalue of p(A)
6. Diagonalize a matrix.
Surprisingly no question on singular value decomposition! Not hard, although I didn’t 4b and only partially did 4c. But tomorrow’s complex analysis, also at 0930, will be a bit harder. Shit.
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