Suppose $\mathbf{u}$ and $\mathbf{v}$ are the first two columns of a 3 by 3 matrix $A$. Which third columns
$\mathbf{w}$ would make this matrix singular? Describe a typical column picture of $A\mathbf{x} = \mathbf{b}$ in that singular case, and a typical row picture (for a random $\mathbf{b}$).$\boxed{\text{(P38) Column picture:}}$ $A\mathbf{x} = \mathbf{b}$ asks for a combination of columns to produce $\mathbf{b}$.
$\boxed{\text{(P38) Row picture:}}$ Each equation in $A\mathbf{x} = \mathbf{b}$ gives a line (n = 2) or a plane (n = 3) or a "hyperplane" (n > 3). They intersect at the solution or solutions, if any.
Answer: $A$ is singular when its third column $\mathbf{w}$ is a combination $c\mathbf{u} + d\mathbf{v}$ of the first columns.
$\color{red}{\Large{[}}$ A typical column picture has $\mathbf{b}$ outside the plane of $\mathbf{u, v, w}. \color{red}{\Large{]}}$ $\color{#0070FF}{\Large{[}}$ A typical row picture has the intersection line of two planes parallel to the third plane. $\color{#0070FF}{\Large{]}}$ Then no solution.
"Singularity" hasn't been formally defined, but P27 identifies it with dependent columns. So the third column depends on the first two, ie: $\mathbf{w} =c\mathbf{u} + d\mathbf{v} \; \forall \; c, d \in \mathbb{R}$. Moreover, $\mathbf{w}$ is a plane on which the vectors $\mathbf{u, v}$ lie. I seize this paragraph.
$\large{1.}$ How and why is the red bracket true? $\qquad \large{2.}$ How and why is the blue bracket true?
Please mind that this question is from but Section 2.1 of IoLA, 4th ed, by Strang, so please keep answers rudimentary.
Best Answer
Your first question: the red bracket is true because b is randomly chosen, it can be either on the plane or out of the plane of u, v, w. Furthermore, the possibility that it's out of the plane is bigger than that it's on the plane.