Set 1: systems, due Sep 2.
Set 2: matrices, due Sep 6.
Set 3: matrix factorisation, due Sep 11.
Set 4: vector space, span, linear independence, due Oct 2.
Set 5: dimension, coördinates, due Oct 23. Many of the problems ask to
find a basis for some space or subspace. Finding the dimension is an
exercise in counting degrees of freedom. Finding a basis can often be
achieved by guessing, by analogy with the canonical bases for R^n or
for spaces of matrices (see, e.g., example 2 of § 3.4). For some
problems, like problem 7, the approach is to require the entries of
the matrix to satisfy a certain linear system; a basis is then
obtained by finding the solutions in parametric form - each parameter
multiplies one basis element.
Set 6: transformations I, due Oct 30.
Set 7: transformations II, due Nov 6.
Set 8: scalar product, due Nov 13.
Set 9: orthogonality, due Nov 20.
Set 10: least squares, inner product, due Nov 29.
Set 11: eigenvalues, due Dec 4.