Due: October 30th, 2017

Math 18 Assignment 4

Answers to these problems are now available here.

  1. Determine which of the following matrices are invertible. Find the inverse of those which are.
    1. $\begin{bmatrix}1&2&0\\0&1&2\\3&9&-1\\4&-4&-2\end{bmatrix}$
    2. $\begin{bmatrix}6\end{bmatrix}$
    3. $\begin{bmatrix}9&8&7\\6&5&4\\3&2&1\end{bmatrix}$
    4. $\begin{bmatrix}0&1&-1\\-1&0&1\\2&-2&0\end{bmatrix}$
  2. Find all values of $k$ for which the following matrix is invertible. $$\begin{bmatrix}k & 0 & 2 \\ 3 & 3 & -2 \\ 3 & 2 & -2 \end{bmatrix}$$
  3. Let $\mathbb{P}$ be the set of polynomials with real coefficients. Which of the following sets are vector subspaces of $\mathbb{P}$?
    1. $W_1 = \left\{p \in \mathbb{P} : p(4) = 0\right\}$
    2. $W_2 = \left\{p \in \mathbb{P} : p(0) = 4\right\}$
    3. $W_3 = \left\{p \in \mathbb{P} : p \text{ has only terms of even degree}\right\}$
    Note that, for example, $p(x) = -x + 4$ is in both $W_1$ and $W_2$ since $p(0) = 4$ and $p(4) = 0$. However, $p \notin W_3$ since it has a term of odd degree, namely $-x$. On the other hand, $q(x) = 3x^4 - 6x^2 + 2 \in W_3$.
  4. None of the following are vector spaces. In each case, show that one of the properties of vector space is not satisfied.
    1. The integer lattice $\mathbb{Z}^3 := \left\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \text{ are integers}\right\}$, with the usual multiplication and addition.
    2. The following set of points in the plane: $\left\{(x, y) \in \mathbb{R}^2 : xy \geq 0\right\}$, together with the usual multiplication and addition.
    3. The set of strings of zero or more letters, where "$+$" corresponds to concatenation and for any number $\lambda \in \mathbb{R}$ and any string of letters $\vec{w}$, $\lambda\vec{w} = \vec{w}$.

      For example, if $\vec{v} = \textrm{''}snow\textrm{''}$ and $\vec{w} = \textrm{''}flake\textrm{''}$, then $\vec{v} + \vec{w} = \textrm{''}snowflake\textrm{''}$. Similarly, $3\vec{v} = \textrm{''}snow\textrm{''}$. If $\vec{\varepsilon}$ is the string containing zero letters, and $\vec{x}$ is another string, then $\vec{\varepsilon} + \vec{x} = \vec{x} = \vec{x} + \vec{\varepsilon}$. So for example, $\textrm{''}\textrm{''} + \textrm{''}snow\textrm{''} = \textrm{''}snow\textrm{''} = \textrm{''}snow\textrm{''} + \textrm{''}\textrm{''}$.

  5. Find a spanning set for the null space of each of the following matrices.
    1. $\begin{bmatrix}1&4&7\end{bmatrix}$
    2. $\begin{bmatrix}0&1&3&0&2&5\\0&0&0&1&-2&4\\0&0&0&0&0&0\end{bmatrix}$
    3. $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}$
    4. $\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$
  6. Suppose that $T: \mathbb{R}^3 \to \mathbb{R}^7$ is a linear transformation, and $\begin{bmatrix}3\\-2\\1\end{bmatrix} \in \mathrm{ker}(T)$. Show that there is a non-zero linear map $S : \mathbb{R^2} \to \mathbb{R^3}$ so that $T(S(\vec{x})) = \vec{0}$ for all $\vec{x} \in \mathbb{R}^2$.