AP® Precalculus · Unit 4 · Topic 4.12
Linear Transformations of the Plane
A linear transformation uses a matrix to move every point in the plane at once, sending each vector to a weighted combination of the matrix's columns. It always fixes the origin in place.
A 2x2 matrix is a machine for moving points. Feed it a vector (x, y), and it hands back a new vector: x times the first column, plus y times the second column. Since this rule applies to every point in the plane simultaneously, a whole picture gets stretched, squished, rotated, or flipped as one piece. Straight lines stay straight, evenly spaced points stay evenly spaced, and the grid pattern of the plane gets warped into a new grid.
That column rule is the mechanism behind everything. If a matrix has columns (a, c) and (b, d), the vector (1, 0) always lands on the first column (a, c), and (0, 1) always lands on the second column (b, d). Any other vector (x, y) is just a mix of those two basic vectors, so its image is x(a, c) + y(b, d). This is why the columns of a matrix tell you the whole story: they are the images of the two simplest building blocks of the plane.
This sits right next to vectors and matrix arithmetic in the unit because a linear transformation is really matrix-vector multiplication viewed as a picture. It also sets up why translation gets excluded from this family: shifting everything by a fixed amount moves the origin, but every linear transformation sends (0, 0) to (0, 0), since any matrix times the zero vector gives zero. That single fact is the cleanest test for whether a transformation can be written as a matrix at all.
Key terms
- linear transformation
- A rule from a matrix that sends each vector to a combination of the matrix's columns, fixing the origin.
- column
- A vertical list of numbers in a matrix; each column shows where a basic unit vector lands.
- unit square
- The square with corners (0,0), (1,0), (1,1), and (0,1), often used to visualize a transformation's effect.
- diagonal matrix
- A matrix with nonzero entries only on the main diagonal, which scales x and y independently.
Watch it explained
- How it worksHow does a linear transformation send each input vector to a combination of the columns?
- Worked exampleShow what the matrix with rows (2, 0) and (0, 3) does to the unit square and to the vector (1, 1).
- Going furtherWhat happens to the origin under any linear transformation at all?
- Common mistakeWhy isn't sliding the whole plane over by 3 a linear transformation?
Check yourself
End of topic.