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

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.