Question

In the link bellow at time 10:40 in the the video of this first lecture, he starts to draw the "column picture" or the vectors composed of the columns of the array matrix.

This is conceptually confusing in that he's graphing the vectors in two dimensional space but both values in each vector represent an X coredinate or a y coardinate for the two equations we started with. What axis are these being graphed on?

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations

Answer 1

Many people have the same type of question when first learning linear algebra. It is important to be able to look at an x and y graph as merely a visualization tool. A function can be visually represented in many ways. The column space is another of these ways.

Consider a two by two matrix. The matrix can be transposed making the rows columns and the columns rows. Has anything really changed? Did we lose any information? Nope. Therefore, what is stopping us from transposing a matrix and graphing it "sideways", if we wanted to?

The way Prof. Strang is visualizing the column space is as vectors rather than a line. I suggest looking at the x and y graph as a \[x_1\] and \[x_2\] graph. Horizontal values are x1, and vertical values are x2. Ah, so now we look at the x value of the first row as x1, and the x value from the second row as x2. Now we can visualize a vector <x1,x2>.