$\newcommand{\ones}{\mathbf 1}$
Norm and distance
What is $\|(1,2,3)\|$?
$1$
Incorrect.
$3$
Incorrect.
$\sqrt{14}=3.742$
Correct!
$6$
Incorrect.
$14$
Incorrect.
Suppose $x$ is an $n$-vector, with $n>1$. Then $x^2$
is the inner product of $x$ with itself
Incorrect.
is the vector with entries $x_i^2$
Incorrect.
makes no sense
Correct!
Given a vector $x$
$\|x\| = - \|-x\|$
always
Incorrect.
sometimes
Correct!
when $x = 0$
never
Incorrect.
$\|(x,x)\| = $
$\|x\|$
Incorrect.
$2\|x\|$
Incorrect.
$\sqrt{2} \|x\|$
Correct!
$\frac{1}{2} \|x\|$
Incorrect.
Can we generalize the triangle inequality to three vectors? I.e., do we have $\|a + b + c\| \leq \|a\| + \|b\| + \|c\|$ for any $n$-vectors $a, b, c$?
Yes
Correct!
$\|a + b + c\| \leq \|a + b\| + \|c\| \leq \|a\| + \|b\| + \|c\|$.
No
Incorrect.
Suppose the $n$-vectors $a$ and $b$ are orthogonal. What is $\mathbf{rms}(a+b)$?
$\mathbf{rms}(a)+\mathbf{rms}(b)$
Incorrect.
$\sqrt{\mathbf{rms}(a)^2+\mathbf{rms}(b)^2}$
Correct!
it cannot be determined
Incorrect.
Consider three $n$-vectors $a,b,c$.
$\angle(a,c) = \angle(a,b)+\angle(b,c)$
Incorrect.
$\angle(a,c)$ cannot be determined from $\angle(a,b)$ and $\angle(b,c)$
Correct!
Suppose $x$ is a $20$-vector with $\|x\| = 10$. Which of the statements below follow from the Chebyshev inequality?
no $x_i$ can satisfy $|x_i| \geq 10.5$
✓
✗
This option is correct.
at least one $x_i$ must have magnitude at least $2$
✓
✗
This option is incorrect. This is true, but doesn't follow from the Chebyshev inequality.
no more than $3$ entries of $x$ can exceed $6$ in magnitude
✓
✗
This option is correct.
no more than half the entries of $x$ are positive
✓
✗
This option is incorrect. This is false.
Submit
Which of the following are true?
$\mathbf{std}(x) \leq \mathbf{rms}(x)$
✓
✗
This option is correct.
$\mathbf{rms}(x) \leq |\mathbf{avg}(x)|$
✓
✗
This option is incorrect.
$\mathbf{std}(x)^2 = \mathbf{rms}(x)^2 - \mathbf{avg}(x)^2$
✓
✗
This option is correct.
$\mathbf{std}(x)=0$ only when all entries of $x$ are equal
✓
✗
This option is correct.
Submit
The notation $x \perp y$ means
$x$ and $y$ have equal norm
Incorrect.
$x$ and $y$ are aligned
Incorrect.
$\angle(x,y)=\pi/2$
Correct!
$x$ and $y$ make an obtuse angle
Incorrect.
Suppose $x$ gives the daily temperature in Palo Alto and $y$ gives the daily temperature in the neighboring city Menlo Park, over the same 5 year period. We would expect
$x$ and $y$ are approximately uncorrelated
Incorrect.
$x$ and $y$ are highly correlated
Correct!
Suppose that $\|a+b\|< \|a\|$. Then
$a \perp b$
Incorrect.
$\angle(a,b)>90^\circ$
Correct!
$\|b\|< \|a\|$
Incorrect.
Suppose $x$ and $y$ are Boolean feature vectors (i.e., each entry is either $0$ or $1$) encoding the presence of symptoms in patients Alice and Bob. Which of the following are true statements?
$x^Ty$ is number of symptoms Alice and Bob have in common
✓
✗
This option is correct.
$\|x\|^2$ is number of symptoms Alice has
✓
✗
This option is correct.
$\ones^Ty$ is number of symptoms Bob has
✓
✗
This option is correct.
$\|x-y\|^2$ is number of symptoms Alice has but Bob does not
✓
✗
This option is incorrect.
$\ones^T(x-y)$ is number of symptoms Alice has but Bob does not
✓
✗
This option is incorrect.
$x \perp y$ means that Alice and Bob do not share any symptoms
✓
✗
This option is correct.
Submit
Assets AAA and BBB have daily returns over three days given by $a=(+0.01,-0.01,+0.03)$ and $b=(-0.01,0.00,+0.02)$. Which of the following are correct?
AAA has higher (mean) return than BBB
✓
✗
This option is correct.
BBB has lower risk than AAA
✓
✗
This option is correct.
AAA is a better investment than BBB
✓
✗
This option is incorrect.
AAA and BBB are likely in the same industry sector
✓
✗
This option is incorrect.
Submit