Due: November 15th, 2017

Math 18 Assignment 6

Answers to these problems are now available here.

  1. Consider the space of polynomials of degree at most $4$, $\mathbb{P}_4$. The standard basis for $\mathbb{P}_4$ is given by the monomials: $\mathcal{E} = \{1,x,x^2,x^3,x^4\}$. However, there are other bases for the polynomials which may be more convenient to work with in some settings; two are detailed below.

    The Chebyshev polynomials (of the second type) are useful in approximation theory and polynomial interpolation, and are defined by the conditions $U_0(x) = 1$, $U_1(x) = 2x$, and for $n > 1$, $U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$. In particular, the first five Chebyshev polynomials are as follows: $$\mathcal{B} = \{1, 2x, 4x^2-1, 8x^3-4x, 16x^4-12x^2+1\}.$$

    The Hermite polynomials likewise have many uses, being related to the behaviour of a quantum harmonic oscillator and random matrix theory. Their general form is not as easily presented, but the first five Hermite polynomials are the following: $$\mathcal{C} = \{1, x, x^2-1, x^3-3x, x^4-6x^2+3\}.$$

    Compute the following change of basis matrices.

    1. $[I]_{\mathcal{B}}^{\mathcal{E}}$, the change of basis matrix from $\mathcal{B}$-coordinates to $\mathcal{E}$-coordinates.
    2. $[I]_{\mathcal{C}}^{\mathcal{E}}$, the change of basis matrix from $\mathcal{C}$-coordinates to $\mathcal{E}$-coordinates.
    3. $[I]_{\mathcal{C}}^{\mathcal{B}}$, the change of basis matrix from $\mathcal{C}$-coordinates to $\mathcal{B}$-coordinates.

  2. Suppose that $V$ is a vector space, and $\mathcal{B}, \mathcal{C}$, and $\mathcal{D}$ are bases of $V$. Furher, suppose that the following change of basis matrices, converting from $\mathcal{B}$-coordinates to $\mathcal{C}$- and $\mathcal{D}$-coordinates, are known: $$[I]_{\mathcal{B}}^{\mathcal{C}} = \begin{bmatrix} -2 & -1 & -1 \\ -3 & 1 & -1 \\ -2 & 1 & 0 \end{bmatrix} \hspace{2in} [I]_{\mathcal{B}}^{\mathcal{D}} = \begin{bmatrix} -2 & -4 & 2 \\ 2 & -1 & -2 \\ 2 & -2 & 4 \end{bmatrix}.$$
    1. What is $\dim(V)$? How do you know?
    2. Suppose that $\vec{x} \in V$, and $[\vec{x}]_{\mathcal{B}} = \begin{bmatrix}2 \\ -1 \\ 2\end{bmatrix}$. What is $[\vec{x}]_{\mathcal{C}}$?
    3. Suppose that $\vec{y} \in V$, and $[\vec{y}]_{\mathcal{D}} = \begin{bmatrix}1 \\ -3 \\ -1\end{bmatrix}$. What is $[\vec{y}]_{\mathcal{B}}$?
    4. Suppose that $\vec{z} \in V$, and $[\vec{z}]_{\mathcal{D}} = \begin{bmatrix}20 \\ 0 \\ 5\end{bmatrix}$. What is $[\vec{z}]_{\mathcal{C}}$?
    1. Give an example of two $3\times3$ matrices $A$ and $B$ so that $\det(A+B) \neq \det(A) + \det(B)$.
    2. Give an example of a $3\times 3$ matrix $A$ and a scalar $c \in \mathbb{R}$ so that $\det(cA) \neq c\det(A)$.
  3. Let $A$ be the following $3\times3$ matrix: $$A = \begin{bmatrix} -11/2 & 5 & -5/2 \\ -3/2 & 3 & -3/2 \\ 2 & 2 & -1 \end{bmatrix}.$$ For which values of $k$ is the following matrix invertible? $$A - kI_3 = \begin{bmatrix} -11/2 - k & 5 & -5/2 \\ -3/2 & 3 - k & -3/2 \\ 2 & 2 & -1 - k \end{bmatrix}$$
  4. An $n\times n$ matrix is said to be orthogonal if $AA^T = A^TA = I_n$; that is, if $A^T = A^{-1}$. (It turns out that in two or three dimensions, the orthogonal matrices describe rotations and reflections.) Suppose that $T : \mathbb{R}^3 \to \mathbb{R}^3$ is a linear transformation with standard matrix $A$, and $A$ is orthogonal. Show that $T$ preserves volume: if $S \subset \mathbb{R}^3$ is a solid with volume $V$, then $T(S)$ also has volume $V$.