Cheatsheet: Undoing a transformation
Systems are transformations
Section titled “Systems are transformations”A linear system is M · v = b: find the input v that the transformation M sends to the target b.
The inverse
Section titled “The inverse”M^-1 undoes M. Defined by:
M^-1 · M = M · M^-1 = I I = [ 1 0 ] (the do-nothing transformation) [ 0 1 ]When it exists, the system solves as v = M^-1 · b (unique solution).
It exists exactly when det(M) != 0. Zero determinant means a collapse, information lost, no undo possible.
The three spaces and numbers
Section titled “The three spaces and numbers”| Idea | What it is | Where it lives |
|---|---|---|
| Column space | All possible outputs = span of the columns | Output side |
| Rank | Dimension of the column space (how many output dims are used) | A number |
| Null space | All inputs crushed to zero (M·v = 0) | Input side |
| Nullity | Dimension of the null space | A number |
Conservation (rank-nullity): rank + nullity = input dimension. Dimensions kept + dimensions lost = dimensions started with.
Three equivalent invertibility tests
Section titled “Three equivalent invertibility tests”All true together, or all false together:
det(M) != 0- Full rank (column space = entire input dimension)
- Null space =
{0}(only the origin gets crushed)
Worked examples
Section titled “Worked examples”| Matrix | det | Column space / rank | Null space / nullity | Invertible? |
|---|---|---|---|---|
[[2,1],[1,1]] | 1 | all of 2D, rank 2 | {0}, nullity 0 | Yes |
[[2,4],[1,2]] | 0 | line through [2,1], rank 1 | line through [-2,1], nullity 1 | No |
Both satisfy rank + nullity = 2.
Solving M·v = b
Section titled “Solving M·v = b”| Matrix | Outcome |
|---|---|
| Invertible | Exactly one solution (v = M^-1·b) |
Not invertible, b off the column space | No solution (target unreachable) |
Not invertible, b on the column space | Infinitely many (one solution + any null-space vector) |
Why it matters for AI
Section titled “Why it matters for AI”- Rank = how many dimensions a matrix actually uses. Big matrices are often low-rank, hence compressible.
- Null space = directions the transformation is blind to (maps to the same place). Lost information, named.
Pitfalls to dodge
Section titled “Pitfalls to dodge”- Inverse is “one over the matrix.” No. It is the undo transformation;
M^-1·M = I, not division. - Every system has a solution. No. Non-invertible gives none or infinitely many.
- Column space vs null space. Outputs reachable vs inputs destroyed. Different spaces.
- Rank = matrix size. No. Rank is output dimensions used; a big matrix can have small rank.
The one-line version
Section titled “The one-line version”The inverse undoes a transformation when nothing was lost; the determinant, the rank, and the null space are three ways of checking whether anything was.