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. In particular, the determinant of an upper triangular matrix is the product of the diagonal entries.

Additional Problems

The following problems are taken from S. Roman, Advanced Linear Algebra (3rd ed.), Exercises for Chapter 2, pp. 72–73 (numbered as in the text). Note that \mathcal{L}(V,W) is the space of linear maps from V to W.

  1. Let A\in\mathcal{M}_{m,n} have rank k. Prove that there are matrices X\in\mathcal{M}_{m,k} and Y\in\mathcal{M}_{k,n}, both of rank k, for which A=XY. Prove that A has rank 1 if and only if it has the form A=x^ty where x and y are row matrices.

  2. Let \tau\in\mathcal{L}(V,W). Prove that \tau is an isomorphism if and only if it carries a basis for V to a basis for W.

  3. Let \mathcal{T} be a subset of \mathcal{L}(V). A subspace S of V is \mathcal{T}-invariant if S is \tau-invariant for every \tau\in\mathcal{T}. Also, V is \mathcal{T}-irreducible if the only \mathcal{T}-invariant subspaces of V are \{0\} and V. Prove the following form of Schur’s lemma. Suppose that \mathcal{T}_V\subseteq\mathcal{L}(V) and \mathcal{T}_W\subseteq\mathcal{L}(W) and V is \mathcal{T}_V-irreducible and W is \mathcal{T}_W-irreducible. Let \alpha\in\mathcal{L}(V,W) satisfy \alpha\tau_V=\tau_W\alpha, that is, for any \mu\in\mathcal{T}_V there is a \lambda\in\mathcal{T}_W such that \alpha\mu=\lambda\alpha. Prove that \alpha=0 or \alpha is an isomorphism.

  4. Let S be a subspace of V. Show that there is a \tau\in\mathcal{L}(V) for which \ker(\tau)=S. Show also that there exists a \sigma\in\mathcal{L}(V) for which \mathrm{im}(\sigma)=S.

  5. Let A be an m\times n matrix over F. What is the relationship between the linear transformation \tau_A:F^n\to F^m and the system of equations AX=B? Use your knowledge of linear transformations to state and prove various results concerning the system AX=B, especially when B=0.