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📘 Lesson 7 of 7 · Linear Algebra

🔄 Linear Transformations

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.

Course progress: 100%

01 Key Concepts

Linear 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.

Matrix 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.

Common Transformations

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

Rotation Matrix

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

Composition of Transformations

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

02 Key Formulas

03 Solved Examples

Example 1 A transformation is defined by matrix A=[[2,0],[0,2]]. Find T(v) for v=(3,4).
  1. Apply T(v) = A*v: [[2,0],[0,2]] * (3,4).
  2. Top: 2*3+0*4=6. Bottom: 0*3+2*4=8.
Answer: T(v) = (6,8) -- this is a scaling transformation that doubles every vector
Example 2 A transformation is defined by matrix A=[[0,-1],[1,0]]. Find T(v) for v=(1,0).
  1. Apply T(v)=A*v: [[0,-1],[1,0]] * (1,0).
  2. Top: 0*1+(-1)*0=0. Bottom: 1*1+0*0=1.
Answer: T(v) = (0,1) -- this matrix rotates vectors 90° counterclockwise
Example 3 Verify that T(v)=A*v is linear by checking T(u+v)=T(u)+T(v) conceptually.
  1. Matrix multiplication distributes over addition: A*(u+v) = A*u + A*v, which is a basic property of matrix algebra.
  2. This confirms the additive property holds for any matrix-based transformation.
Answer: Confirmed: matrix transformations are always linear, since matrix multiplication distributes over vector addition

04 Practice Questions

1A=[[3,0],[0,3]]. Find T(v) for v=(2,5).
(6,15)
2A=[[1,0],[0,-1]]. Find T(v) for v=(4,7). (This reflects over the x-axis)
(4,-7)
3What matrix represents a 90° counterclockwise rotation?
[[0,-1],[1,0]]
4What are the two defining properties of a linear transformation?
T(u+v)=T(u)+T(v), and T(c*v)=c*T(v)
5What kind of transformation does A=[[2,0],[0,2]] represent?
A uniform scaling (doubling) transformation

📄 Linear Transformations — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet