Matrix Representation of Linear Maps: A Simple Guide

Every linear map between finite-dimensional spaces can be written as a matrix, but the matrix is not the map: it is the map translated into the language of two chosen bases. Fix ordered bases B in the domain and C in the codomain; the matrix records, column by column, the C-coordinates of the images of the B basis vectors.

This translation makes linear algebra computable. Composition becomes matrix multiplication, invertibility becomes matrix inversion, and basis change becomes conjugation. Building the matrix correctly, column by column, is the skill unlocking all of it.

Key Concepts

• Matrix of T relative to B, C: column j is [T(bj)]C, the C-coordinates of T(bj).
• Fundamental identity: [T(v)]C = [T]B to C [v]B, converting B-coordinates of input to C-coordinates of output.
• Standard matrix: with standard bases, columns are simply T(e1), T(e2), and so on.
• Shape: for T from n-dimensional to m-dimensional space, the matrix is m by n.

Formulas

• Column rule: column j = [T(bj)]C.
• Action: [T(v)]C = [T]B to C [v]B.
• Composition: [S after T] = [S][T], right to left.
• Change of basis: for an operator, the new-basis matrix is P inverse [T] P.
• Inverse map: [T inverse] = ([T]) inverse.

Worked Example

Let T(x,y) = (3x+y, x minus 2y) on R squared with standard bases. T(1,0) = (3,1) and T(0,1) = (1,minus 2), so A = [[3,1],[1,minus 2]]. Test on v = (2,5): formula gives (6+5, 2 minus 10) = (11,minus 8); matrix multiplication gives (3(2)+5, 2 minus 2(5)) = (11,minus 8), agreeing. Composition check: T(T(1,0)) = T(3,1) = (10,1). And A squared = [[10,1],[1,5]], whose first column (10,1) matches, confirming that applying T twice is A squared.

Common Misconceptions

• Entries are the values of T. Entries are coordinates of T(bj) in the C basis; with nonstandard bases you must expand each image in C.
• Composition multiplies left to right. It is right to left: S after T is (matrix of S)(matrix of T).
• Similar matrices are different maps. P inverse A P is the same operator in a different basis.

Real-World Applications

• Graphics: model, view and projection matrices are chained linear maps multiplied each frame.
• Robotics: each joint contributes a matrix; the end-effector pose is their product.
• Neural networks: every dense layer is a matrix of a linear map plus a nonlinearity.

Feed each domain basis vector through T, expand in the codomain basis, stack as columns: that is the matrix. Then coordinates transform by multiplication, composition by product, and the abstract map is fully computable.