Week 3 Suggested Problems

Suggested Problems (suggested by Claude)

  1. Let L:\mathbf{R}^3\to\mathbf{R}^3 be the linear map with matrix [L] = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{bmatrix} and let W=\mathrm{Span}\{e_1,e_2\}, which is L-invariant. Compute \mathrm{trace}(L), \mathrm{trace}(L_W), and \mathrm{trace}(\overline{L}) on \mathbf{R}^3/W, and check that \mathrm{trace}(L)=\mathrm{trace}(L_W)+\mathrm{trace}(\overline{L}).

  2. Using cofactor expansion, compute \det(A) for A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \end{bmatrix}, then find the adjugate A^{\mathrm{adj}} and verify that AA^{\mathrm{adj}}=\det(A)I.

  3. Use Cramer’s rule to solve \begin{aligned} x + y + z &= 2\\ 2x - y + z &= 3\\ x + 3y - z &= -1 \end{aligned} and check your answer by another method (e.g. row reduction).

Suggested Problems (suggested by JT)

  1. Suppose V is a vector space and W\subset V is a subspace. Suppose that L:V\to V is a linear map and W is L-invariant, meaning L(W)\subset W. Let L_{W}:W\to W be the restriction of L to W.

In this case we have a well defined linear map \overline{L}:V/W\to V/W defined by \overline{L}([v])=[L(v)].

I claim in the notes that \det(L)=\det(L_W)\det(\overline{L}).

The idea is that one can choose a basis for W, extend it to a basis for V, and then the matrix A of L in that basis has a block form with a matrix B for L_{W} in the upper left and C for \overline{L} in the lower left.

Then I claim that \det(A)=\det(B)\det(C).

Check these claims more carefully. In particular, verify that the bases exist, that the matrices have the claimed form, and that the determinant of a “block triangular” matrix is in fact the product of the determinants of the blocks.