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Linear Algebra

Homework 5 - Vector spaces, subspaces, and spanning sets

Due 2021-09-28.

  1. Find a spanning set for the null space of the matrix \(A = \begin{pmatrix} -1 & 2 & -3 & 4 & 2\\ 3 & -2 & 1 & -4 & -1\\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}\)

  2. Determine if the following are subspaces of $\operatorname{Func}(\mathbb{R},\mathbb{R})$ or not:
    • The set of odd functions. Recall that $f$ is odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$.
    • The set of solutions to the differential equation \(\frac{d^2f}{dx^2} + 3 \frac{df}{dx} + 2 f = e^x.\)
    • The set of injective functions.
    • The set of surjective functions.

  3. Find the LU factorization, with partial pivoting, of \(A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{pmatrix}\) over both $\mathbb{Q}$ and $\mathbb{F}_2$.

  4. Prove the Proposition from the notes that describes spanning sets for null spaces of matrices.

    • Give a set that is closed under scalar multiplication but not addition.
    • Give a set that is closed under addition but not scalar multiplication.
    • Give a set that is closed under neither.