Chapter 3 · Structure

Vector Spaces and Linear Transformations

One question

What does a matrix keep, collapse, and reach?

Before the code

Four ideas to keep in view

Follow an input through the matrix and ask what survives at the output.

01Reachability: $b \in \operatorname{Col}(A)$
A solution exists exactly when b is reachable from the columns of A.
02Uniqueness: $\operatorname{Null}(A) = \{0\}$
If a solution exists, it is unique exactly when no nonzero input direction disappears.
03Dimension: $\operatorname{rank}(A) + \operatorname{nullity}(A) = n$
For $A \in \mathbb R^{m\times n}$ , the number of pivots plus the number of free variables equals n, the number of input coordinates.
04Linear map: $A(cx+y)=cAx+Ay$
A matrix preserves vector addition and scalar multiplication.

Structure bridge

Different inputs can reach the same output

For $A = [[1,2],[2,4]]$ , the nonzero direction $z = (-2,1)^{\mathsf T}$ disappears. Adding z to an input cannot change the output. Each row in the display below lists one vector’s coordinates.

Inputs · x and x + z

01z = (−2, 1)ᵀa null direction
02Az = 0the direction disappears
03A(x + z) = Axthe output stays fixed

Outputs · same b

Chapter 3Null space controls uniqueness; column space controls existence.

Practice sequence

Four ideas to keep in view

Different inputs can reach the same output

Choose a lab