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

Vectors

$a = (4, -2, 0)$

Which of the following is true?

$(a, a)$
  1. is invalid notation
    Incorrect.
  2. is a $6$-vector
    Correct!
  3. is a $3$-vector
    Incorrect.

$a - [4, -2, 0]$
  1. is invalid notation
    Correct!
  2. is $0$
    Incorrect.
  3. is not $0$
    Incorrect.

$a - \left[ \begin{array}{c} 4\\-2\\3 \end{array}\right]$
  1. is invalid notation
    Incorrect.
  2. is $0$
    Incorrect.
  3. is not $0$
    Correct!

$a + (2, 6)$
  1. doesn't make sense
    Correct!
  2. is $(6, 4, 3)$
    Incorrect.
  3. is $(4, -2, 3, 2, 6)$
    Incorrect.

What is \[ 2 \left[ \begin{array}{c} 1\\ -1 \\ 2 \end{array}\right] -\left[ \begin{array}{c} 2\\ 0 \\ -1 \end{array}\right]? \]
  1. $(0,-2,3)$
    Incorrect.
  2. $(0,-2,5)$
    Correct!
  3. $(0,-2,1)$
    Incorrect.

The $n$-vector $c$ represents the daily earnings of a company over $n$ days (with negative entries meaning a loss on that day).

The number $\textbf{1}^T c$ represents
  1. the last day's earnings
    Incorrect.
  2. the average earnings over the $n$ days
    Incorrect.
  3. the total earnings over the $n$ days
    Correct!
  4. the difference between the first day's and last day's earnings
    Incorrect.

Given $n$-vectors $a, b, x$, and scalar $\alpha$.

If $a^Tb = 0$, then
  1. at least one of $a$ or $b$ must be $0$
    Incorrect.
  2. both $a$ and $b$ must be $0$
    Incorrect.
  3. both $a$ and $b$ can be nonzero
    Correct!

If $\alpha x = 0$, then
  1. one of $\alpha$ or $x$ must be $0$
    Correct!
  2. both $\alpha$ and $x$ must be $0$
    Incorrect.
  3. both $\alpha$ and $x$ can be nonzero
    Incorrect.

A particular computer can compute the inner product of two $10^6$-vectors in around $0.001$ second. The same computer can compute the inner product of two $10^7$-vectors in (approximately) how long?
  1. $0.001$ second
    Incorrect.
  2. $0.01$ second
    Correct!
  3. $0.1$ second
    Incorrect.
  4. $1$ second
    Incorrect.

The regression model $\hat y = x^T\beta +v$ predicts the life span (age at death) of a person in some population, where the feature vector $x$ encodes various attributes of the person. Assuming the model fits actual life span data reasonably well (although of course not very accurately for any particular individual) what would you guess about $\beta_3$, if $x_3=1$ means the person is a smoker, and $x_3=0$ means the person is not a smoker?
  1. you can't say without knowing what the other features are
    Incorrect.
  2. $\beta_3$ is likely positive
    Incorrect.
  3. $\beta_3$ is probably small
    Incorrect.
  4. $\beta_3$ is likely negative
    Correct!
  5. $\beta_3$ is larger in magnitude than the other $\beta_i$'s
    Incorrect.