$\newcommand{\ones}{\mathbf 1}$

# Linear independence

The following questions refer to the vectors $a = \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix},\quad b = \begin{bmatrix} 2 \\ 6 \\ 4 \end{bmatrix},\quad c = \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix},\quad d = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix}.$

Are $a$ and $b$ linearly independent?
1. Yes
Incorrect. $a = \frac{1}{2} b$, so $a$ and $b$ are not linearly independent.
2. No
Correct!

Can $a$ be written as a linear combination of $c$ and $d$?
1. Yes
Correct!
2. No
Incorrect. $a$ can be written as a linear combination of $c$ and $d$: $a = c + \frac{1}{2}d$

Are $b$, $c$, and $d$ linearly independent?
1. Yes
Incorrect.
2. No
Correct!

What is the angle between $c$ and $d$?
1. Acute
Incorrect.
2. $90^{\circ}$
Correct!
3. Obtuse
Incorrect.

Suppose $q_1$, $q_2$, and $q_3$ are orthonormal $n$-vectors, and $x$ is an $n$-vector. Then
1. $n \geq 3$
Correct!
2. $n = 3$
Incorrect.
3. $n < 3$
Incorrect.

$x = (q_1^Tx)q_1 + (q_2^Tx)q_2 + (q_3^Tx)q_3$
1. is always true
Incorrect.
2. is never true
Incorrect.
3. is true when $n=3$
Correct!
4. is true when $n>3$
Incorrect.

The Gram-Schmidt algorithm determines if a set of $n$ $n$-vectors is
1. mutually acute
Incorrect.
2. nonzero
Incorrect.
3. a basis
Correct!

The Gram-Schmidt algorithm (or equivalent variations, like the modified Gram-Schmidt algorithm)
1. is illegal in several states
Incorrect.
2. should always be carried out by hand
Incorrect.
3. is readily carried out on a laptop computer on a list of 100 vectors of dimension $10000$
Correct!
4. is a challenge even for modern, fast computers
Incorrect.
5. requires considerable skill to carry out, which however can be learned
Incorrect.