Linear algebra - Coordinate Systems I'm preparing for an upcoming Linear Algebra exam, and I have come across a question that goes as follows: Let U = {(s, s − t, 2s + 3t)}, where s and t are any real numbers. Find the coordinates of x = (3, 4, 3) relative to the basis B if x is in U . Sketch the set U in the xyz-coordinate system.
It seems that in order to solve this problem, I'll have to find the basis B first! how do I find that as well?
The teacher hae barely covered the coordinate systems and said she will less likely include anything from the section on the exam, but I still want to be safe. The book isn't of much help. It explains the topic but doesn't give any examples.
Another part of the question ask about proving that U is a subspace of R3, but I was able to figure that one out on my own. I'd appreciate if someone could show me how to go about solving the question above.

This might be an answer, depending on how one interprets the phrase "basis $B$", which is undefined in the question as stated:
Note that $(s, s - t, 2s + 3t) = s(1, 1, 2) + t(0, -1, 3)$.  Taking $s = 1$, $t = 0$ shows that $(1, 1, 2) \in U$.  Likewise, taking $s = 0$, $t = 1$ shows $(0, -1, 3) \in U$ as well.  Incidentally, the vectors $(1, 1, 2)$ and $(0, -1, 3)$ are clearly linearly independent; to see this in detail, note that if  $(s, s - t, 2s + 3t) = s(1, 1, 2) + t(0, -1, 3) = (0,0,0)$, then we must obviously have $s = 0$, whence $s - t = -t = 0$ as well.  Assuming $B$ refers to the basis $(1, 1, 2)$, $(0, -1, 3)$ of $U$, it is easy to work out the values of $s$ and $t$ corresponding to $x$:  setting $(s, s - t, 2s + 3t) = (3, 4, 3)$, we see that we must have
$s = 3$ whence $t = -1$ follows from $s - t = 4$.  These check against $2s + 3t = 3$, as the reader may easily verify.  The desired coordinates for $x$ in the basis $B$ are thus $(3, -1)$.
Think that about covers it, if my assumption about $B$ is correct.  
Can't provide a graphic, but one is easily constructed noting that the vectors $(1, 1, 2)$, $(0, -1, 3)$ span $U$ in $R^3$ (the "$xyz$" coordinate system).
