Linear Algebra · Quick Reference

Linear Transformations Cheat Sheet

Linear Algebra · Lesson 7/7

In one line: a linear transformation is a function between vector spaces that preserves addition and scalar multiplication — and every one of them can be represented by a matrix.

Key Ideas

1Linear Transformation Definition. A function T that satisfies T(u+v) = T(u)+T(v) and T(c*v) = c*T(v) for all vectors u,v and scalars c.

2Matrix Representation. Every linear transformation from one finite-dimensional space to another can be represented by multiplying input vectors by a matrix: T(v) = A*v.

3Common Transformations. Scaling (stretching/shrinking), rotation (turning around the origin), reflection (flipping across a line), and shearing (slanting) are all linear transformations.

4Rotation Matrix. Rotating a 2D vector by angle theta counterclockwise uses the matrix [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]].

5Composition of Transformations. Applying one linear transformation after another corresponds to multiplying their matrices together (in the correct order).

Worked Examples

A transformation is defined by matrix A=[[2,0],[0,2]]. Find T(v) for v=(3,4).

T(v) = (6,8) -- this is a scaling transformation that doubles every vector

A transformation is defined by matrix A=[[0,-1],[1,0]]. Find T(v) for v=(1,0).

T(v) = (0,1) -- this matrix rotates vectors 90° counterclockwise

Verify that T(v)=A*v is linear by checking T(u+v)=T(u)+T(v) conceptually.

Confirmed: matrix transformations are always linear, since matrix multiplication distributes over vector addition

Formulas

T(v) = A*v

Rotation matrix: [[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]

Practice Yourself

A=[[3,0],[0,3]]. Find T(v) for v=(2,5).

(6,15)

A=[[1,0],[0,-1]]. Find T(v) for v=(4,7). (This reflects over the x-axis)

(4,-7)

What matrix represents a 90° counterclockwise rotation?

[[0,-1],[1,0]]