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

Norm and distance

What is $\|(1,2,3)\|$?
  1. $1$
    Incorrect.
  2. $3$
    Incorrect.
  3. $\sqrt{14}=3.742$
    Correct!
  4. $6$
    Incorrect.
  5. $14$
    Incorrect.

Suppose $x$ is an $n$-vector, with $n>1$. Then $x^2$
  1. is the inner product of $x$ with itself
    Incorrect.
  2. is the vector with entries $x_i^2$
    Incorrect.
  3. makes no sense
    Correct!

Given a vector $x$

$\|x\| = - \|-x\|$
  1. always
    Incorrect.
  2. sometimes
    Correct! when $x = 0$
  3. never
    Incorrect.

$\|(x,x)\| = $
  1. $\|x\|$
    Incorrect.
  2. $2\|x\|$
    Incorrect.
  3. $\sqrt{2} \|x\|$
    Correct!
  4. $\frac{1}{2} \|x\|$
    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 $a, b, c$?
  1. Yes
    Correct! $\|a + b + c\| \leq \|a + b\| + \|c\| \leq \|a\| + \|b\| + \|c\|$.
  2. No
    Incorrect.

Suppose the $n$-vectors $a$ and $b$ are orthogonal. What is $\mathbf{rms}(a+b)$?
  1. $\mathbf{rms}(a)+\mathbf{rms}(b)$
    Incorrect.
  2. $\sqrt{\mathbf{rms}(a)^2+\mathbf{rms}(b)^2}$
    Correct!
  3. it cannot be determined
    Incorrect.

Consider three $n$-vectors $a,b,c$.
  1. $\angle(a,c) = \angle(a,b)+\angle(b,c)$
    Incorrect.
  2. $\angle(a,c)$ 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?

Which of the following are true?

The notation $x \perp y$ means
  1. $x$ and $y$ have equal norm
    Incorrect.
  2. $x$ and $y$ are aligned
    Incorrect.
  3. $\angle(x,y)=\pi/2$
    Correct!
  4. $x$ 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
  1. $x$ and $y$ are approximately uncorrelated
    Incorrect.
  2. $x$ and $y$ are highly correlated
    Correct!

Suppose that $\|a+b\|< \|a\|$. Then
  1. $a \perp b$
    Incorrect.
  2. $\angle(a,b)>90^\circ$
    Correct!
  3. $\|b\|< \|a\|$
    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?

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?