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Syllabus [tex]


Exams
Exam 1: [pdf] [tex]
Exam 2: [pdf] [tex]
Exam 3: [pdf] [tex]


Homework
Homework 1 (due 21 August) (solution): Section 1.1: #11, 12, 17, 18, 31, 32, 33, 34, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 52, 53, 54, 55, 57 (a,b)

Homework 2 (due 28 August) (solution): Section 2.1: 1, 4, 5, 6, 11, 12, 27, 28, 29, 30; Section 2.2: 1, 2, 3, 4, 9 (only do part b), 10 (only do part b), 25, 26, 32; Section 2.3: 1, 2, 3, 4, 7, 8, 9, 10, 22, 23, 24, 25, and the following additional problems:
Problem A: Solve the system of equations (field: $\mathbb{C}$) $$\left\{ \begin{array}{llll} (2+i)z_1 & + 3z_2 &= 1 \\ - z_1 &+ iz_2 &= 0 \\ \end{array} \right.$$ note: it is helpful to remember that to "simplify" a complex number of the form $\dfrac{a+bi}{c+di}$ you may multiply by a "convenient form of $1$" defined from the "complex conjugate" of the denominator as follows: $$\dfrac{a+bi}{c+di} = \dfrac{a+bi}{c+di} \dfrac{c-di}{c-di} = \dfrac{(a+bi)(c-di)}{c^2+d^2} = \dfrac{ac + bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i.$$ Problem B: Solve the system of equations (field: $\mathbb{Z}_5$) $$\left\{ \begin{array}{lll} 3x_1 &+ 4x_2 &= 1 \\ x_1 &- 2x_2 &= 0 \end{array} \right.$$
Homework 3 (due 6 September) (solution): Section 2.3: 1, 2, 3, 4, 7, 8, 9, 10, 22, 23, 24, 25, 30, 31; Section 3.1: #1, 2, 3, 4 and the following two problems:
Problem A: Determine if the following set of vectors is linearly independent (field: $\mathbb{C}$): $$\left\{ \left[ \begin{array}{ll} 3 \\ -i \\ 2 \end{array} \right], \left[ \begin{array}{ll} 2 \\ i \\ 3 \end{array} \right], \left[ \begin{array}{ll} 1 \\ 1 \\ 1 \end{array} \right] \right\}$$ Problem B: Determine if the following set of vectors is linearly independent (field: $\mathbb{Z}_3$): $$\left\{ \left[ \begin{array}{ll} 1 \\ 2 \\ \end{array} \right], \left[ \begin{array}{ll} 2 \\ 1 \end{array} \right] \right\}$$ Problem C: Determine if the following set of vectors is linearly independent (field: $\mathbb{Z}_5$): $$\left\{ \left[ \begin{array}{ll} 3 \\ 2 \\ 0 \end{array} \right], \left[ \begin{array}{ll} 1 \\ 1 \\ 1 \end{array} \right], \left[ \begin{array}{ll} 1 \\ 2 \\ 1 \end{array} \right] \right\}$$
Homework 4 (due 11 September) (solution): Section 3.1: #5, 6, 7, 8, 9, 10, 17, 22, 36, 40; Section 3.2: #3, 4, 6, 7, 14, 15, 44(b) and the following additional problems:
Problem A: If $A \in \mathbb{Z}_5^{2 \times 2}$ and $B \in \mathbb{Z}_5^{2 \times 2}$ with $$A = \left[ \begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array} \right]$$ and $$B = \left[ \begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array} \right],$$ compute $AB, BA, A^TB^T, B^TA^T,$ and $(BA)^T$. Are any of these matrices related?

Problem B: If $A, B, C \in \mathbb{C}^{2 \times 2}$ with $$A = \left[ \begin{array}{ll} 1 & 1 \\ i & 0 \end{array} \right],$$ $$B = \left[ \begin{array}{ll} 0 & 2+i \\ 1 & 1 \end{array} \right],$$ and $$C = \left[ \begin{array}{ll} -1 & 1+i \\ 1-i & 1 \end{array} \right].$$ Use the definition of linear independence (i.e. the equation) and solve it to determine whether $\{A,B,C\}$ is a linearly independent or linearly dependent set of matrices.
Homework 5 (due 25 September) (solution): see this file

Homework 6 (due 29 September) (solution): Section 3.3: #2, 3, 4, 50, 54 ;Section 6.1 #2, 8, 10, 14, 16, 18, 20, 24, 26, 28, 34, 36

Homework 7 (due 4 October) (solution): Section 6.1: #46,48, 50, 53, 54, 61, 62

Homework 8 (due 11 October) (solution): Section 6.2: #5, 6, 7, 8, 10, 11, 18, 22

Homework 9 (due 18 October) (solution): Section 6.2: #19, 23, 27, 28, 29, 34, 35, 39, 44 (proof by contradiction!)

Homework 10 (due 23 October) (solution): Section 6.3: #1, 2, 6, 7, 9; Section 6.4: #1, 2, 3, 5, 6, 9, 10, 14, 17, 24, 27

Homework 11 (due 30 October) (solution): Section 6.4: #20, 25; Section 6.5: #2, 3, 4, 9, 11, 15, 16, 21, 25, 28, 32

Homework 12 (due 8 November) (solution): Section 6.5: #15, 16, 21, 25, 28, 32; and the following additional problem:

Problem A: ("Laguerre polynomials") Let $T_n \colon \mathscr{P}_2 \rightarrow \mathscr{P}_2$ be defined by $T_n(p)=xp''+(1-x)p'+np$ for $n=0,1,2$. Find $\mathrm{ker}(T_n)$ for $n=0,1,2$.

Homework 13 (due 15 November) (solution): The following homework refers to Examples in the notes on inner products.

Problem A. Let $H=(\mathbb{R}^3,\langle \cdot,\cdot \rangle)$ be the inner product space of Example 1. Let $\vec{x}=\begin{bmatrix} 3 \\ 2\\ -1\end{bmatrix}$ and $\vec{y}=\begin{bmatrix} 2 \\ 1 \\ 17\end{bmatrix}$. Compute $\langle \vec{x},\vec{y} \rangle$.
Problem B. Let $H=(\mathbb{P}, \langle \cdot,\cdot \rangle)$ be the inner product space of Example 2. Let $\vec{p}(x)=x-1$, $\vec{q}(x)=x^2$. Compute both of the inner products $\langle \vec{p},\vec{q} \rangle$ and $\langle \vec{p},\vec{p} \rangle$ using integration by parts.
Problem C. Let $H=(\mathscr{C}[0,1],\langle \cdot,\cdot \rangle)$ be the inner product space of Example 3. Let $f(x)=\log(x+1)$ and $g(x)=1$. Calculate $\langle f,g \rangle$. Let $h_1(x)=x^2$ and $h_2(x)=\sin(x)$. Calculate $\langle h_1,h_2 \rangle$ (hint: use integration by parts or Wolframalpha).
Problem D. Let $H=(\ell^1(\mathbb{R}),\langle\cdot,\cdot\rangle)$ be the inner product space of Example 4. Let $\{a_k\}_{k=0}^{\infty} = \left\{ \dfrac{1}{3^k} \right\}_{k=0}^{\infty}$ and $\{b_k\}_{k=0}^{\infty} = \left\{ \dfrac{1}{7^k} \right\}_{k=0}^{\infty}$. Calculate $\left\langle \left\{ a_k \right\}, \left\{b_k\right\} \right\rangle$ (hint: this is a geometric series). Let $\left\{c_k\right\}=\left\{d_k\right\}=\sqrt{\dfrac{1}{k!}}$. Calculate $\langle c_k,d_k \rangle$ (hint:recall the power series $e^x = \displaystyle\sum_{k=0}^{\infty} \dfrac{x^k}{k!}$).
Problem E. Let $H=(\mathbb{C},\langle\cdot,\cdot\rangle)$ be the inner product space of Example 5. Let $\vec{x}=5+4i$ and $\vec{y}=9-11i$. Compute $\langle \vec{x},\vec{y}\rangle$. Let $z_1=21+16i$ and $z_2=\dfrac{11-5i}{2+i}$. Calculate $\langle z_1,z_2\rangle$ (hint: mutltiply $z_2$ by $1=\dfrac{2-i}{2-i}$ to put $z_2$ into the form $z_2=a+bi$; this is similar to "rationalizing denominators").
Problem F.: Let $(\mathbb{R}^{4 \times 1},\langle \vec{x},\vec{y} \rangle)$ be an inner product space where $\langle \vec{x},\vec{y} \rangle$ denotes dot product.
Show that the vectors $\vec{a}=\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$ and $\vec{b}=\begin{bmatrix} -4 \\ -3 \\ 2 \\ 1 \end{bmatrix}$ are orthogonal vectors.
Problem G.: Show that set $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$ is a mutually orthogonal set of vectors. Also show that the set $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$ is not a mutually orthogonal set of vectors.
Problem H.: Consider the vector space $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where the inner product is given by $$\langle p(x),q(x) \rangle = \displaystyle\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2} dx.$$ It can be shown (via methods of calculus 2) that the moments in this inner product space are $$\langle 1,1 \rangle=\sqrt{\pi},$$ $$\langle x,1 \rangle=0,$$ $$\langle x^2,1 \rangle=\dfrac{\sqrt{\pi}}{2},$$ $$\langle x^3,1 \rangle = 0,$$ $$\langle x^4,1 \rangle = \dfrac{3\sqrt{\pi}}{4},$$ $$\langle x^5,1 \rangle = 0,$$ $$\langle x^6,1 \rangle = \dfrac{15\sqrt{\pi}}{8}.$$ Use these moments and the "linear in the first argument" property of inner products to compute $\langle 4x^2+3x+9,1\rangle$ and $\langle 32x^5-64x^3+24x,1\rangle$.
Problem I.: Consider the inner product space $(C[0,1],\langle \cdot,\cdot \rangle)$ where $$\langle f,g \rangle = \displaystyle\int_0^1 f(x)g(x)x^2 dx.$$ Compute $\mathrm{proj}_{x^2-3x} (5x+2)$ and $\mathrm{proj}_{5x+2}(x^2-3x)$.
Problem J.: Consider the inner product space $(\mathbb{R}^{3\times 1},\langle \cdot,\cdot \rangle)$, where $\langle \cdot,\cdot \rangle$ denotes the dot-product. Consider the set $\{v_1,v_2,v_3\}$ where $v_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \vec{v}_2 = \begin{bmatrix} 2 \\ 4 \\ 8 \end{bmatrix}, \vec{v}_3 = \begin{bmatrix} 3 \\ 9 \\ 27 \end{bmatrix}$. It is clear that this set is not an orthogonal set of vectors. Apply the Gram-Schmidt process to orthogonalize this set.
Problem K.: Consider the inner product space $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where $\langle \vec{p},\vec{q} \rangle = \displaystyle\int_{-1}^1 \vec{p}(x)\vec{q}(x) dx$. Apply the Gram-Schmidt process to the sequence $(x^n)_{n=0}^{\infty}$ to find the first four polynomials orthogonal with respect to $\langle \cdot,\cdot \rangle$. (note: these polynomials are called Legendre polynomials)

Additional notes
1. Notes on inner product spaces
2. Notes on orthogonality
3. Notes on Gram-Schmidt
External links
  1. "Essence of Linear Algebra" by 3blue1brown
  2. Linear algebra video lectures from MIT
  3. Linear algebra video lectures from Princeton
  4. "How Google converted language translation into a problem of vector space mathematics" (this paper is referenced)
  5. What's linear about linear algebra?