Due: November 6th, 2017

Math 18 Assignment 5

Answers to these problems are now available here.

  1. Convince yourself that you are comfortable solving problems similar to 5-26 (except 17(e)) in section 4.6 of the text. No submission is necessary.
  2. For each of the following pairs of matrices, the second is the reduced row echelon form of the first. Find a basis for the column space, row space, and null space of the first, non-row-reduced matrix. Then state its rank.
    1. $$\begin{bmatrix}-4&-1&-2&4&-2\\1&1&1&5&1\\-3&-3&-3&3&-4\end{bmatrix}\hspace{2in}\begin{bmatrix}1&0&1/3&0&1/6\\0&1&2/3&0&10/9\\0&0&0&1&-1/18\end{bmatrix}$$
    2. $$\begin{bmatrix}2&0&3&1\\4&5&7&1\\-2&-5&-4&2\\4&5&7&-5\end{bmatrix}\hspace{2in}\begin{bmatrix}1&0&3/2&0\\0&1&1/5&0\\0&0&0&1\\0&0&0&0\end{bmatrix}$$.
  3. Let us consider the vector space $\mathbb{P}_2$, of polynomials of degree at most $2$. Notice that the set $Q$ of polynomials with no linear term (i.e., those of the form $ax^2 + c$) is a subspace of $\mathbb{P}_2$. It turns out that any such polynomial $ax^2 + c \in Q$ can be written as a linear combination as follows: $$ax^2 + c = a(x^2 + x) - c(2x - 1) + (2c-a)x.$$ Is $\{x^2+x, 2x - 1, x\}$ a basis for $Q$? Why or why not?
  4. For each of the following subspaces, determine its dimension and provide a basis.
    1. $\left\{\begin{bmatrix}s+2t+3r\\4s+5t+6r\\7s+8t+9r\end{bmatrix} : s,t,r \in \mathbb{R}\right\} \subseteq \mathbb{R}^3.$
    2. $\left\{\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix} \in \mathbb{R}^4 : x_1 + x_2 + x_3 + x_4 = 0\right\} \subseteq \mathbb{R}^4.$
  5. Suppose that $W \subseteq \mathbb{R}^n$ has a basis $\{\vec{v}_1, \ldots, \vec{v}_k\}$. Suppose further that $\vec{v}_{k+1}, \ldots, \vec{v}_{n}$ are chosen so that $\{\vec{v}_1, \ldots, \vec{v}_n\}$ is a basis for $\mathbb{R}^n$. Show that there is a linear map $T : \mathbb{R}^n \to \mathbb{R}^{\color{red}{n-k}}$ so that $\ker(T) = W$.
  6. Consider the space of polynomials $\mathbb{P}$. Let $D : \mathbb{P} \to \mathbb{P}$ and $S : \mathbb{P} \to\mathbb{P}$ be linear maps such that for all $k \in \mathbb{N}$, $D(x^k) = kx^{k-1}$ and $S(x^k) = \frac1{k+1}x^{k+1}$.

    1. Show that $D$ is onto but not one-to-one. (Hint: to show onto, you should show that the image of $D$ contains a basis for $\mathbb{P}$; why is this enough to conclude that $D$ is onto?)
    2. Show that $S$ is one-to-one but not onto. (Hint: show that $D\circ S$ is one-to-one first, and use this to conclude that $S$ itself must be one-to-one.)

    We know that linear maps between finite dimensional vector spaces are one-to-one if and only if they are onto (and that this occurs exactly when the standard matrix of such a map is invertible). What we have shown here is that this does not hold for vector spaces of infinite dimension.

  7. Given a finite set $S = \{s_1, \ldots, s_k\}$, the free vector space on $S$ is the set of formal linear combinations of elements of $S$: $$\mathcal{F}(S) = \left\{a_1s_1 + \cdots + a_ks_k : a_1, \ldots, a_k \in \mathbb{R}\right\}.$$ Addition is defined by collecting like terms, and scalar multiplication multiplies the weight of each element. $\mathcal{F}(S)$ should be thought of the vector space obtained by declaring $S$ to be a basis.

    Consider the free vector space on the set $Q = \{|0\rangle, |1\rangle\}$; then $\mathcal{F}(Q)$ is a 2-dimensional vector space, and all vectors are of the form $\alpha|0\rangle + \beta|1\rangle$. Define the vectors $|+\rangle$ and $|-\rangle$ by $$|+\rangle = \frac1{\sqrt2}|0\rangle + \frac1{\sqrt2}|1\rangle \hspace{2in} |-\rangle = \frac1{\sqrt2}|0\rangle - \frac1{\sqrt2}|1\rangle.$$

    1. Show that $\{|+\rangle, |-\rangle\}$ is a basis for $\mathcal{F}(Q)$.
    2. Let $\mathcal{B}$ be the ordered basis $\{|+\rangle, |-\rangle\}$. Find $\left[|0\rangle\right]_{\mathcal{B}},$ $\left[|1\rangle\right]_{\mathcal{B}},$ and $\left[\frac35|0\rangle + \frac45|1\rangle\right]_{\mathcal{B}}$.

    It turns out that in the vector space $\mathcal{F}_{\mathbb{C}}(Q)$ (which is like $\mathcal{F}(Q)$ with complex coefficients rather than real coefficients), the vectors $\alpha|0\rangle + \beta|1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$ describe the possible states of a qubit, or quantum bit. If $W$ is the set $\{|w\rangle : w \text{ is a binary string of length } k\}$, then the state of a quantum system with $k$ qubits can be described as the vectors in $\mathcal{F}_{\mathbb{C}}(W)$ with coefficients whose square absolute values sum to $1$. A quantum computer acts on such a quantum state by applying linear maps with the property that every quantum state is sent to another state (so, for example, such maps must have trivial kernel). There's a lot of fascinating stuff here which is well beyond the scope of this course; Wikipedia is a good place to start reading.