Link Search Menu Expand Document
Linear Algebra

Homework 13 - Volumes and characteristic polynomials

  1. Find the volume of the parallelopiped spanned by the vectors \(\begin{pmatrix} 1 \\ 1 \\ -1 \\ 1 \end{pmatrix}, \ \begin{pmatrix} 0 \\ -1 \\ -1 \\ 0 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}\)

  2. Let $A$ be an orthogonal $n \times n$ matrix. What is the volume of the image of the unit hybercube $[0,1]^n$ under $A$?

  3. Let $A$ and $B$ by $n \times n$ matrices. Assume that $B = CAC^{-1}$. Show that \(\chi_A(x) = \chi_B(x)\)

  4. Find the eigenspaces of the matrix \(\begin{pmatrix} 3 & 1 \\ -2 & 1 \end{pmatrix}\)

  5. Suppose is $A$ is a block diagonal matrix with blocks $A_1, \ldots, A_l$. What is the relation between the eigenspaces of $A$ and the eigensapces of the $A_i$?