Linear transformations & matrices
The idea
A linear transformation is a function between vector spaces that respects the structure: T(u + v) = T(u) + T(v) and T(cv) = c T(v). Rotations, reflections, scalings, projections, and shears all qualify; translations do not, because they move the origin. The decisive theorem of the subject: a linear map from Rⁿ to Rᵐ is completely determined by where it sends the standard basis vectors, and stacking those images as columns produces the standard matrix A, after which applying T is just matrix-vector multiplication.
Think of a linear map as a controlled distortion of space: grid lines stay parallel and evenly spaced, and the origin stays put. Because every vector is a combination of basis vectors, linearity propagates the basis images to everything — two function values (in R²) pin down the entire map. The common misconception is that any formula with x and y is linear; T(x, y) = (x + 1, y) and T(x, y) = (x², y) both fail, the first because T(0, 0) ≠ (0, 0), the second because doubling the input does not double the output.
Worked example
Let T: R² → R² be the linear transformation T(x, y) = (2x + y, x − 3y). Find the standard matrix A of T, and use it to compute T(4, −1).
- Find the images of the standard basis: T(1, 0) = (2, 1) and T(0, 1) = (1, −3) — these two outputs encode the entire transformation.
- Assemble A by columns: the first column is T(1, 0) and the second is T(0, 1), so A has first row (2, 1) and second row (1, −3).
- Apply A to the vector (4, −1) by taking row-by-column dot products: first entry 2(4) + 1(−1) = 8 − 1 = 7, second entry 1(4) + (−3)(−1) = 4 + 3 = 7.
- So T(4, −1) = (7, 7). Cross-check straight from the formula: 2(4) + (−1) = 7 and 4 − 3(−1) = 7 — the matrix and the formula agree, as they must.
- Confirm linearity is doing the work: (4, −1) = 4e₁ − e₂, so T(4, −1) = 4T(e₁) − T(e₂) = 4(2, 1) − (1, −3) = (8 − 1, 4 + 3) = (7, 7), the same answer assembled from basis images alone.
Answer. The standard matrix has rows (2, 1) and (1, −3), and T(4, −1) = (7, 7).
Check your understanding
- Why does knowing T on the standard basis vectors determine T on every vector in the space?
- How can you tell quickly that a translation like T(x, y) = (x + 1, y) cannot be linear?
- What does the composition of two linear maps correspond to at the matrix level, and why does order matter?
- How would you read off from the columns of a matrix what the transformation does to the unit square?
Build the foundations first
Linear transformations & matrices builds on these concepts. If any feel shaky, start there.