Consider the system of equations:
$$
u + v + w = b_1 \\
2u + 3w = b_2 \\
3u + v + 4w = b_3 \\
$$
In section 1 of Gilbert Strang's Linear Algebra and its Applications, there's the following figure (Figure 1.6):
What does it mean for columns to be inside a plane? I know that the columns would be the coefficient vectors for $u, v, w$, or $[1,2,3], [1,0,1], [1,3,4]$. All of these vectors have the same first value of $1$, so I suppose they do reside in the same plane. Therefore, does the figure imply that it is a slice of 3-dimensional space where the first dimension is fixed at 1?
Best Answer
He means a plane through the origin. That’s what the dot at the intersection of the rays representing the vectors in the figure is supposed to convey. It’s certain that not just any plane is meant, since every set of three points is coplanar.