Linear Transformations

A linear map preserves addition and scaling; matrices are its coordinate form, with image and kernel as the key outputs.

The idea

A linear transformation is a map $T$ that sends vectors to vectors and respects addition and scaling: $T(u + v) = T(u) + T(v), \qquad T(cu) = cT(u)$ for all vectors $u, v$ and all scalars $c$. Adding two vectors and then applying $T$ gives the same result as applying $T$ to each and adding the images. In the figure, $T$ carries the parallelogram on $u$ and $v$ to the parallelogram on $T(u)$ and $T(v)$, and the corner $u + v$ lands on $T(u) + T(v)$ — the first rule made visible.

These two rules pin the map down almost completely. Every vector is a combination of a few basis vectors, and linearity pushes $T$ through the combination, so the images of the basis vectors determine the image of every vector. Recording those images as the columns of a matrix $A$ turns the map into $T(x) = Ax$: matrices are how linear maps are written in coordinates.

Two subspaces track what a linear map does: the kernel, the set of vectors sent to $0$, and the image, the set of vectors of the form $T(x)$.

Ways to work on it

• Walkthrough. What linearity means and why kernel/image matter.
• Practice. Apply a small matrix to a vector.
• Hardest. Find a nonzero kernel vector and interpret what it says about the map.

Not sure where to start? Take the ten-question placement test.