Question

Hi. Just saw the video of Prof. Strang's first lecture. Could someone please explain how, in the "column picture", one of the two x coefficients for a vector (say v1) is taken as the x coordinate and the other x coefficient - from the second row or equation - is taken as the y coordinate? Thanks!

Answer 1

you should not mix the x and y coordinates and the 'x' and 'y' of the equation. Later in the course I guess the distinction will be made explicite.
In the mean time understand that the 'x' and 'y' letters of the equation are simply unknown scalars (numbers). We could have call them 'e' and 'f'. In that case Prof. Strang's linear system would be written:
\[2e - f = 0\]\[-e + 2f = 3\] or
\[e \left(\begin{matrix}2 \\ -1\end{matrix}\right)+f \left(\begin{matrix}-1 \\ 2\end{matrix}\right)=\left(\begin{matrix}0 \\ 3\end{matrix}\right)\]
The idea is to find the numbers 'e' and 'f' such that the equation is true.

To visualize the solution, Prof. Strang drew the two vectors (2,-1) and (-1,2).
When you draw a vector in the plane,usualy the first coordinate goes on the x axis and the second on the y axis.

That's what he did for (2,-1) and (-1,2).
Then he added the "right amount" of each vectors to make the solution vector (0,3).

The "right amount" is 'e or 1' of the vector (2,-1) and 'f or 2' of the vector (-1,2).