$\newcommand{\ones}{\mathbf 1}$
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?
Yes
Incorrect.
$a = \frac{1}{2} b$, so $a$ and $b$ are not linearly independent.
No
Correct!
Can $a$ be written as a linear combination of $c$ and $d$?
Yes
Correct!
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?
Yes
Incorrect.
No
Correct!
What is the angle between $c$ and $d$?
Acute
Incorrect.
$90^{\circ}$
Correct!
Obtuse
Incorrect.
Suppose $q_1$, $q_2$, and $q_3$ are orthonormal $n$-vectors, and $x$ is an $n$-vector. Then
$n \geq 3$
Correct!
$n = 3$
Incorrect.
$n < 3$
Incorrect.
$x = (q_1^Tx)q_1 + (q_2^Tx)q_2 + (q_3^Tx)q_3$
is always true
Incorrect.
is never true
Incorrect.
is true when $n=3$
Correct!
is true when $n>3$
Incorrect.
The Gram-Schmidt algorithm determines if a set of $n$ $n$-vectors is
mutually acute
Incorrect.
nonzero
Incorrect.
a basis
Correct!
The Gram-Schmidt algorithm (or equivalent variations, like the modified Gram-Schmidt algorithm)
is illegal in several states
Incorrect.
should always be carried out by hand
Incorrect.
is readily carried out on a laptop computer on a list of 100 vectors of dimension $10000$
Correct!
is a challenge even for modern, fast computers
Incorrect.
requires considerable skill to carry out, which however can be learned
Incorrect.