Cheatsheet: Linear transformations as moves
What makes a transformation linear
Section titled “What makes a transformation linear”A transformation is a function from vectors to vectors (picture every point moving at once). It is linear when both hold:
| Requirement | Meaning |
|---|---|
| Origin stays fixed | The zero vector maps to the zero vector. |
| Lines stay straight | Grid lines stay parallel and evenly spaced; no curving, no bunching. |
Allowed: stretch, rotate, shear, flip. Not allowed: curve, bend, shift the whole plane.
The basis-vector trick
Section titled “The basis-vector trick”A linear transformation is fully determined by where it sends i-hat and j-hat.
v = x · i-hat + y · j-hat ===> L(v) = x · L(i-hat) + y · L(j-hat)Same scalars x, y (the coordinates of v); only the two vectors they scale have changed. Track two landing spots, reconstruct every vector.
A matrix is the record of where the basis landed
Section titled “A matrix is the record of where the basis landed”[ a b ] first column [a, c] = L(i-hat)[ c d ] second column [b, d] = L(j-hat)Matrix-vector product (apply the transformation):
M · v = x · (first column) + y · (second column)This is the linear combination from Lesson 2, with the transformed basis vectors as ingredients. Not a memorized rule.
Worked examples (applied to [3, 4])
Section titled “Worked examples (applied to [3, 4])”| Transformation | Matrix | i-hat lands | j-hat lands | [3, 4] becomes |
|---|---|---|---|---|
| Horizontal stretch x2 | [[2, 0], [0, 1]] | [2, 0] | [0, 1] | [6, 4] |
| 90 deg counterclockwise | [[0, -1], [1, 0]] | [0, 1] | [-1, 0] | [-4, 3] |
Stretch: 3·[2,0] + 4·[0,1] = [6, 4]. Rotation: 3·[0,1] + 4·[-1,0] = [-4, 3].
Sketching a 2x2 matrix’s effect
Section titled “Sketching a 2x2 matrix’s effect”- Read the two columns: they are where
i-hatandj-hatland. - Plot those two image vectors from the origin.
- Draw the parallelogram they span. That is the image of the unit square.
Square stays a square (rotation), becomes a rectangle (axis stretch), becomes a slanted parallelogram (shear), or squashes nearly flat (columns nearly parallel, plane collapsing toward a line).
Pitfalls to dodge
Section titled “Pitfalls to dodge”- Reading rows instead of columns. Meaning lives in the columns: two destination vectors side by side.
- Forgetting the origin is pinned. A fixed shift of the whole plane is affine, not linear; a 2x2 matrix cannot do it.
- Letting the grid curve. If lines bend, it is not a linear transformation.
- Memorizing M·v. It is just
x·(col 1) + y·(col 2).
Words to use precisely
Section titled “Words to use precisely”- Linear transformation: a vector-to-vector function fixing the origin and keeping lines straight, parallel, evenly spaced.
- Matrix: the two transformed basis vectors written as columns.
- Matrix-vector product: reassembling a vector from the transformed basis,
x·col1 + y·col2. - Unit square: corners at origin,
i-hat,j-hat,i-hat + j-hat; its image is the parallelogram of the columns.
The one-line version
Section titled “The one-line version”A matrix is not a grid of numbers; it is a record of where the basis vectors land, and that record moves all of space.