Questions About Column Vectors in A System of Linear Equations

Question

I am a university student who have never touched Linear Algebra before, but I did learn a little bit about matrix and vector from my high school maths class. Now I am trying to learn this subject from watching Gilbert Strang's lecture video on MIT OCW and also reading his book.

While the lecture video is fine, the book is not so easy to understand. For example, I do not understand what question (2) in the picture is trying to present.

After reading the solution, my first thought is that y is perpendicular to all of the three column vectors in A. (dot product=0) Can I say that all the three vectors lie on the same plane as they are all perpendicular to the vector y. And from that, we can conclude that Ax = b has no solution because b is not on that plane (it is not perpendicular to y), can we?

I don't know whether my thought is logical. These all vectors stuff are always harder than they seem. :(

Answer 1

You're correct that the column vectors of $A$ are all perpendicular to $y$, moreover, $b$ is not perpendicular to $y$, what this shows, is that $b$ does not live in the plane spanned by the columns of $A$ (it is a plane, and not all of $\mathbb{R}^3$ precisely because this system has no solution). The plane happens to be a plane that has $y$ as a normal vector. This is related to the concept of the rank of a matrix which is the dimension of the subspace spanned by the columns of the matrix (or equivalently, by the rows of the matrix).

Answer 2

Formally, you can argue that the dot product is bilinear, thus if $y \cdot b = 1$, then also $y \cdot Ax = 1$. Let $x = (x_1,x_2,x_3)$ and let $a_1, a_2, a_3$ be the column vectors of $A$, however you get:

$$y \cdot Ax = y \cdot (x_1a_1 + x_2a_2 + x_3a_3) = \underbrace{y \cdot x_1a_1}_{=0} + \underbrace{y \cdot x_2a_2}_{=0} + \underbrace{y \cdot x_3a_3}_{=0} = 0$$