4.11 Matrices as Functions

Matrices Transform Points

A matrix can act like a function, taking an input vector and mapping it to an output vector through matrix multiplication. Multiplying a 2x2 matrix by a coordinate vector (x, y) produces a new vector (x', y'), effectively transforming a point in the plane. This connects matrices to geometric transformations like rotations, reflections, dilations, and shears.

Composing Transformations

Applying two transformations in sequence corresponds to matrix multiplication of the two matrices (in reverse order of application). If matrix B is applied first, then matrix A, the combined transformation is represented by A·B. Since matrix multiplication is generally not commutative, the order in which transformations are applied matters.

Identity and Inverse Transformations

The identity matrix leaves every vector unchanged, acting like the identity function. If a matrix transformation has an inverse matrix, applying it undoes the original transformation, returning the output vector to its original input. A matrix has an inverse only when its determinant is nonzero.

Key terms

Matrix as a function

A matrix maps an input vector to a unique output vector via matrix multiplication, behaving like a function.

Composition of transformations

Applying B then A corresponds to the matrix product A·B (order matters).

Identity matrix

A matrix that, when multiplied by any vector, leaves it unchanged — acts like the identity function.

Inverse matrix

A matrix that undoes another matrix's transformation; exists only if the determinant is nonzero.

Why order matters in transformations

Matrix multiplication is generally not commutative, so A·B ≠ B·A, meaning the sequence of transformations affects the result.

Try an AP-style question

Applying transformation B and then transformation A corresponds to which matrix?

1. AB
2. BA
3. A + B
4. A − B