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

Linear functions

$a = (4, -2, 0),\quad b = (8, -4, 0)$

Suppose we have a linear function $f:\textbf{R}^3 \rightarrow \textbf{R}$. If $f(a) = 5$, then $f(b)$
  1. $=-5$
    Incorrect.
  2. $=5$
    Incorrect.
  3. $=10$
    Correct!
  4. cannot be determined
    Incorrect.

The $n$-vector $c$ represents the daily earnings of a company over $n$ days, with negative entries meaning a loss on that day. Suppose that $n$ is even, and consider the average earnings in the second half of the period (i.e., over days $i=n/2+1,\ldots, n$) minus the average earnings over the first half of the period (i.e., over days $i=1, \ldots, n/2$). (This quantity gives a kind of trend in average earnings.) Is this a linear function of $c$?
  1. yes
    Correct!
  2. no
    Incorrect.