Let $u$ and $w$ be vectors each having two components and that satisfy

$u + w = \left[ \begin{array}{c} 3 \\ 1 \end{array} \right] \qquad \mbox{and} \qquad u - w = \left[ \begin{array}{c} 1 \\ 3 \end{array} \right]$

Compute $w$. Write your answers as 2 integer values separated by a space.

Compute the cosine of the angle between the vectors

$u = \left[ \begin{array}{c} 3 \\ 4 \end{array} \right] \qquad \mbox{and} \qquad w = \left[ \begin{array}{c} 4 \\ 3 \end{array} \right]$

[Caveat: the question is asking for the cosine of the angle, not the angle itself]

Use one elimination step to make the following system upper triangular. Then use back substitution to solve for $x$ and $y$.

$\begin{array}{r} 2x + 1y = 5 \\ 4x - 2y = 6 \end{array}$

Write your answer as two integer values (the first is $x$, the second $y$) separated by a space.

Compute the matrix product

$\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]$

Write your answer as 4 integer values in column major order (that is $a_{11} \; a_{21} \; a_{12} \; a_{22}$) separated by spaces.

Reduce the following system to upper triangular form and identify the pivots.

$\begin{array}{r} 2x_{1} + 3x_{2} + 1x_{3} = 1 \\ 4x_{1} + 7x_{2} + 5x_{3} = 7 \\ 0x_{1} - 2x_{2} + 2x_{3} = 6 \end{array}$

Write your answer as 3 integer values separated by spaces.

The inverse of the $2 \times 2$ matrix A is

$\big[ A \big]^{-1} = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]^{-1} = k \left[ \begin{array}{rr} d & -b \\ -c & a \end{array} \right]$

where $k$ is a scalar quantity that depends on $a, b, c$, and $d$. Use the identity $A \, A^{-1} = I$ to solve for $k$.

Compute the $L$ and $U$ factors of the coefficient matrix for the following linear system.

$\begin{array}{r} 2x_{1} + 3x_{2} + 1x_{3} = 1 \\ 4x_{1} + 7x_{2} + 5x_{3} = 7 \\ 0x_{1} - 2x_{2} + 2x_{3} = 6 \end{array}$

What is the smallest element in the $L$ matrix?