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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

__External links__
- "Essence of Linear Algebra" by 3blue1brown
- Linear algebra video lectures from MIT
- Linear algebra video lectures from Princeton
- "How Google converted language translation into a problem of vector space mathematics" (this paper is referenced)
- What's linear about linear algebra?