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

# 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.