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 ∈ Col(A)A solution exists exactly when b is reachable from the columns of A.
- 02Uniqueness
Null(A) = {0}A solution is unique when no nonzero input direction disappears.
- 03Dimension
rank(A) + nullity(A) = nThe input space splits between retained and collapsed directions.
- 04Linear map
A(cx + y) = cAx + AyA matrix preserves vector addition and scalar multiplication.
Structure bridge
Different inputs can reach the same output
If a nonzero direction n disappears, adding it to an input cannot change the output.
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11
- 01
n = (−2, 1)a null direction - 02
A n = 0the direction disappears - 03
A(x + n) = Axthe output stays fixed
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Chapter 3Null space controls uniqueness; column space controls existence.
Practice sequence
Choose a lab
- 3.1Finding the Null SpaceTurn one free variable into a direction that the matrix sends to zero.
eliminate → split columns → build a basisOpen lab - 3.2Interpretable Linear TransformationsFollow one vector through rotation, projection, and reflection.
rotate → project → reflectOpen lab - 3.3Solving Non-invertible SystemsCompare a reachable and unreachable target, then trace the full solution family.
test reachability → describe every solutionOpen lab