Algebra → Linear Algebra · Lesson 8

Linear Transformations

Linear transformations are functions between vector spaces that preserve structure. Every matrix IS a linear transformation — and every linear transformation between finite-dimensional spaces IS a matrix.

Key Concepts

Definition

T: ℝⁿ → ℝᵐ is linear if T(u+v) = T(u)+T(v) and T(cv) = cT(v) for all vectors u,v and scalars c. In matrix form: T(x) = Ax where A is the matrix representation.

Standard Transformations in 2D

Rotation by θ: [[cos θ, −sin θ],[sin θ, cos θ]]. Reflection over x-axis: [[1,0],[0,−1]]. Scaling: [[sx,0],[0,sy]]. Shear: [[1,k],[0,1]].

Kernel & Image

Kernel (null space): all x where T(x) = 0. Image (range/column space): all possible outputs T(x). Rank-Nullity: rank(A) + nullity(A) = n (number of columns).

Composition & Inverses

Composing T₁ then T₂: matrix product T₂T₁. Inverse transformation T⁻¹ exists iff A is square with det(A) ≠ 0. T⁻¹T = I (identity).

Live Python Practice

Interactive Lab

Matrix [[a,b],[c,d]]:

Presets:

Gray = original shape, Green = transformed. Blue/red = transformed basis vectors. det = area scale factor.

Check Your Understanding

A linear transformation T satisfies:

The rotation matrix by 90° counterclockwise is:

The kernel of T is:

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