Linear Algebra
3.410 min read

Matrices and Linear Maps: The Equivalence

Every matrix gives a linear map: multiply by the matrix. Every linear map between finite-dimensional spaces is given by multiplication by a matrix. This is the fundamental duality of linear algebra.

Given a linear map T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m, its matrix is formed by computing T(ej)T(\mathbf{e}_j) for each standard basis vector — then assembling the results as columns: A=[T(e1)∣⋯∣T(en)]A = [T(\mathbf{e}_1) \mid \cdots \mid T(\mathbf{e}_n)].

This equivalence means that studying matrices IS studying linear maps. We freely use both languages.

Formal View

Theorem 3.4 — Matrix-Map Equivalence
Every matrix A∈Rm×nA \in \mathbb{R}^{m \times n} defines a linear map TA:Rn→RmT_A: \mathbb{R}^n \to \mathbb{R}^m by TA(x)=AxT_A(\mathbf{x}) = A\mathbf{x}. Conversely, every linear map T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m is equal to TAT_A for a unique matrix AA whose jj-th column is T(ej)T(\mathbf{e}_j).

Why This Matters

This equivalence unifies algebra (matrices) with geometry (transformations), doubling our analytical power.

  • To understand how a matrix transforms space geometrically, think of it as a linear map
  • To compute a linear map efficiently, find its matrix representation
  • Change of basis is a change of matrix representation of the same underlying linear map

Quiz

Question 1

The matrix of the linear map T(x1,x2)=(2x1+x2,3x2)T(x_1, x_2) = (2x_1 + x_2, 3x_2) is:

Common Mistakes

  • Computing columns of AA as T(ej)T(\mathbf{e}_j) but writing entries in the wrong order.
  • Thinking the matrix representation is unique regardless of basis — it depends on the choice of basis for domain and codomain.