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Linear Algebra

Homework 4 - Geometry and Sage

Due 2021-09-23.

  1. For an $m \times n$ matrix $A$, show that the function $\mathbf{x} \mapsto A\mathbf{x}$ is surjective if and only if $\mathcal R(A) = \mathbb{R}^m$.

  2. Recall that $A$ is orthogonal if $A^T = A^{-1}$. Show that $A$ is orthogonal if and only if $f_A$ preserves the scalar product of vectors. Hint: for each index $i$ and $j$ choose $\mathbf{x}$ and $\mathbf{y}$ so that
    \((A\mathbf{x})^T (A\mathbf{y}) = (A^TA)_{ij}\)

  3. Given an example of an orthogonal matrix that is not a permutation matrix.

  4. Write a function in Sage whose input is a matrix of any shape and whose output is the sum of the absolute values of the entries of the matrix.

  5. Use Sage to determine the reduced row echelon form and the LU factorization, with partial pivoting, of the matrix \(A = \begin{pmatrix} 1 & 2 & 4 & 5 & 3 \\ 5 & 4 & 3 & 2 & 3 \\ 1 & -1 & 0 & 1 & 3 \\ 3 & 0 & 1 & 1 & 0 \end{pmatrix}\)