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Syllabus [tex]

Exams
Exam 1 (solution): [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): Section 6.2: #19, 23, 27, 28, 29, 34, 35, 39, 44 (proof by contradiction!)

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