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Matrices and Linear Transformations

This aiMOOC is designed for Grades 11–13 and introduces matrices as both calculation tools and representations of geometric and algebraic transformations. You will learn how vectors can be transformed, how a matrix records what a transformation does to basis vectors, why matrix multiplication represents composition, and how determinants, inverses, and eigenvectors reveal important properties of transformations.

Datei:LinearTransformations.svg

A central idea of this course is that a matrix is more than a rectangular array of numbers. A matrix can describe an action on vectors. When you connect algebraic calculations with geometric movement, stretching, reflection, rotation, and shear, many rules of matrix algebra become easier to understand.

Learning Goals

By the end of this aiMOOC, you should be able to explain the difference between a matrix and a linear transformation, multiply matrices and vectors correctly, construct a matrix from the images of basis vectors, recognize common transformations in the plane, interpret matrix multiplication as composition, use determinants to reason about area scaling and invertibility, calculate simple inverses, and explain the basic meaning of eigenvectors and eigenvalues.

You should also be able to move between three representations: symbolic representation with matrices and formulas, geometric representation with vectors and grids, and verbal representation with precise mathematical explanations.

Foundations: Vectors and Matrices

Vectors as Directed Quantities

A vector can represent displacement, velocity, force, data, or any quantity with several components. In two dimensions, a vector can be written as $𝐯 = (\begin{matrix} x \\ y \end{matrix})$ . The values $x$ and $y$ are the coordinates of the vector relative to a chosen basis.

The standard basis of $ℝ^{2}$ consists of $𝐞_{1} = (\begin{matrix} 1 \\ 0 \end{matrix})$ and $𝐞_{2} = (\begin{matrix} 0 \\ 1 \end{matrix})$ . Every vector $(\begin{matrix} x \\ y \end{matrix})$ can therefore be written as $x 𝐞_{1} + y 𝐞_{2}$ .

Datei:2D basis vectors.svg

This decomposition matters because a linear transformation is completely determined by what it does to the basis vectors. That fact is the bridge between geometric transformations and matrices.

Useful related topics include Vector, Coordinate system, Basis, and Vector space.

Matrix Notation

A matrix is a rectangular arrangement of entries. A general two-by-two matrix is

$A = (\begin{matrix} a & b \\ c & d \end{matrix}) .$

The first number in the size of a matrix is the number of rows, and the second is the number of columns. A matrix with two rows and three columns is therefore a $2 \times 3$ matrix.

For a transformation from $ℝ^{2}$ to $ℝ^{2}$ , a two-by-two matrix is sufficient. For a transformation from $ℝ^{3}$ to $ℝ^{2}$ , a two-by-three matrix is used. In general, an $m \times n$ matrix represents a linear map from $ℝ^{n}$ to $ℝ^{m}$ when the standard bases are used.

Matrix-Vector Multiplication

Suppose

$A = (\begin{matrix} a & b \\ c & d \end{matrix}), 𝐱 = (\begin{matrix} x \\ y \end{matrix}) .$

Then

$A 𝐱 = (\begin{matrix} a x + b y \\ c x + d y \end{matrix}) .$

You can interpret this calculation in two equivalent ways. The row viewpoint uses dot products of rows with the vector. The column viewpoint writes

$A 𝐱 = x (\begin{matrix} a \\ c \end{matrix}) + y (\begin{matrix} b \\ d \end{matrix}) .$

The column viewpoint is especially important for transformations: the first column tells you where $𝐞_{1}$ goes, and the second column tells you where $𝐞_{2}$ goes.

Worked Example: Transforming a Vector

Let

$A = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}), 𝐱 = (\begin{matrix} 3 \\ - 1 \end{matrix}) .$

Then

$A 𝐱 = (\begin{matrix} 2 \cdot 3 + 1 \cdot (- 1) \\ 1 \cdot 3 + 1 \cdot (- 1) \end{matrix}) = (\begin{matrix} 5 \\ 2 \end{matrix}) .$

Geometrically, the vector ending at the point $(3, - 1)$ is sent to the vector ending at $(5, 2)$ . The matrix does not merely store numbers; it defines the rule that creates the new vector.

Linear Transformations

What Makes a Transformation Linear?

A transformation $T$ is linear if it preserves vector addition and scalar multiplication. For all vectors $𝐮$ and $𝐯$ and every scalar $k$ ,

$T (𝐮 + 𝐯) = T (𝐮) + T (𝐯)$

and

$T (k 𝐮) = k T (𝐮)$ .

These two properties imply $T (𝟎) = 𝟎$ . Therefore, every linear transformation sends the origin to the origin.

A rotation about the origin, a reflection in a line through the origin, a scaling from the origin, and a shear are linear transformations. A translation by a nonzero fixed vector is not linear because it moves the origin. A translation can instead be treated as an affine transformation.

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The Matrix of a Linear Transformation

Suppose $T : ℝ^{2} \to ℝ^{2}$ is linear. Because every vector satisfies

$(\begin{matrix} x \\ y \end{matrix}) = x 𝐞_{1} + y 𝐞_{2},$

linearity gives

$T ((\begin{matrix} x \\ y \end{matrix})) = x T (𝐞_{1}) + y T (𝐞_{2}) .$

This means that once you know $T (𝐞_{1})$ and $T (𝐞_{2})$ , you know the transformation of every vector.

If

$T (𝐞_{1}) = (\begin{matrix} p \\ r \end{matrix}) and T (𝐞_{2}) = (\begin{matrix} q \\ s \end{matrix}),$

then the matrix of $T$ in the standard basis is

$A = (\begin{matrix} p & q \\ r & s \end{matrix}) .$

The images of the basis vectors become the columns of the matrix.

Worked Example: Building a Matrix from a Rule

Suppose a transformation sends

$𝐞_{1} \mapsto (\begin{matrix} 1 \\ 2 \end{matrix}) and 𝐞_{2} \mapsto (\begin{matrix} - 1 \\ 3 \end{matrix}) .$

Then

$A = (\begin{matrix} 1 & - 1 \\ 2 & 3 \end{matrix}) .$

For a vector $𝐱 = (\begin{matrix} 4 \\ 2 \end{matrix})$ ,

$A 𝐱 = 4 (\begin{matrix} 1 \\ 2 \end{matrix}) + 2 (\begin{matrix} - 1 \\ 3 \end{matrix}) = (\begin{matrix} 2 \\ 14 \end{matrix}) .$

This example shows why the columns matter. Instead of memorizing a rule, you can reconstruct the entire transformation from the transformed basis.

A Gallery of Common Transformations

Scaling

A scaling transformation changes size. The matrix

$S = (\begin{matrix} s_{x} & 0 \\ 0 & s_{y} \end{matrix})$

scales the horizontal coordinate by $s_{x}$ and the vertical coordinate by $s_{y}$ . If $s_{x} = s_{y}$ , the scaling is uniform. If the factors differ, shapes may stretch differently in different directions.

For example,

$(\begin{matrix} 2 & 0 \\ 0 & \frac{1}{2} \end{matrix})$

doubles horizontal lengths and halves vertical lengths.

Rotation

A counterclockwise rotation through an angle $θ$ about the origin is represented by

$R_{θ} = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) .$

For $θ = 9 0^{\circ}$ ,

$R_{9 0^{\circ}} = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) .$

Thus $(\begin{matrix} 1 \\ 0 \end{matrix})$ is sent to $(\begin{matrix} 0 \\ 1 \end{matrix})$ , exactly as expected for a quarter-turn.

Datei:Rotation map.svg

Rotations preserve distances, angles, and area. Their determinants are $1$ .

Reflection

A reflection in the $x$ -axis is represented by

$(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) .$

It sends $(x, y)$ to $(x, - y)$ . A reflection reverses orientation. In two dimensions, a reflection matrix has determinant $- 1$ .

A reflection in the line $y = x$ is represented by

$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}),$

which exchanges the two coordinates.

Shear

A horizontal shear can be represented by

$H = (\begin{matrix} 1 & k \\ 0 & 1 \end{matrix}) .$

It maps $(x, y)$ to $(x + k y, y)$ . Horizontal lines stay horizontal, but vertical lines generally become slanted.

Datei:Shear.svg

A shear preserves area when its determinant is $1$ , even though it changes angles and usually changes lengths.

Projection

The matrix

$P = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})$

projects every vector in the plane onto the $x$ -axis. Information about the original vertical coordinate is lost. This explains why the transformation cannot be inverted.

Projection is a useful example of a transformation with determinant $0$ : the plane is collapsed onto a line.

Matrix Multiplication as Composition

When one transformation is followed by another, matrix multiplication represents the combined action. If $B$ acts first and $A$ acts second, then

$𝐱 \mapsto B 𝐱 \mapsto A (B 𝐱) = (A B) 𝐱 .$

Therefore, the combined matrix is $A B$ .

Datei:MatrixMultiplication.png

Why Order Matters

Matrix multiplication is generally not commutative: $A B$ is usually different from $B A$ . Geometrically, changing the order of transformations can change the final result.

Consider

$R = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$

for a $9 0^{\circ}$ counterclockwise rotation and

$S = (\begin{matrix} 2 & 0 \\ 0 & 1 \end{matrix})$

for a horizontal stretch.

If you rotate first and stretch second, the combined matrix is

$S R = (\begin{matrix} 0 & - 2 \\ 1 & 0 \end{matrix}) .$

If you stretch first and rotate second, the combined matrix is

$R S = (\begin{matrix} 0 & - 1 \\ 2 & 0 \end{matrix}) .$

Since these products are different, the resulting transformations are different.

How to Multiply Two Matrices

For compatible matrices, each entry of the product is obtained by combining a row of the first matrix with a column of the second matrix.

If

$A = (\begin{matrix} a & b \\ c & d \end{matrix}), B = (\begin{matrix} e & f \\ g & h \end{matrix}),$

then

$A B = (\begin{matrix} a e + b g & a f + b h \\ c e + d g & c f + d h \end{matrix}) .$

The size rule is important: an $m \times n$ matrix can multiply an $n \times p$ matrix, producing an $m \times p$ matrix.

The geometric interpretation helps you remember why the rule is arranged this way: the product must describe how the second transformation changes basis vectors and how the first transformation then acts on those results.

Determinants: Measuring Area Change

For a two-by-two matrix

$A = (\begin{matrix} a & b \\ c & d \end{matrix}),$

the determinant is

$\det (A) = a d - b c .$

The absolute value $| \det (A) |$ tells you the factor by which areas are scaled under the transformation.

Datei:Determinant-as-area.svg

If $| \det (A) | = 3$ , a region of area $2$ is transformed into a region of area $6$ . If $\det (A) = - 3$ , the area scale factor is still $3$ , but orientation is reversed.

Determinant Zero

If $\det (A) = 0$ , two-dimensional area is collapsed to zero. The image may lie on a line or at a point. In this situation, different input vectors can be sent to the same output, so the transformation is not one-to-one and cannot have an inverse.

This connects several ideas:

Geometric view: dimension is collapsed.

Algebraic view: the columns are linearly dependent.

Transformation view: some information is lost.

Inverse view: no inverse matrix exists.

Inverse Transformations

An inverse transformation reverses the effect of a transformation. If $A$ is invertible, its inverse $A^{- 1}$ satisfies

$A^{- 1} A = A A^{- 1} = I,$

where

$I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$

is the identity matrix.

For

$A = (\begin{matrix} a & b \\ c & d \end{matrix})$

with $a d - b c \neq 0$ ,

$A^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix}) .$

For example, if

$A = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}),$

then $\det (A) = 1$ and

$A^{- 1} = (\begin{matrix} 1 & - 1 \\ - 1 & 2 \end{matrix}) .$

You can check the result by multiplying $A A^{- 1}$ and confirming that the result is $I$ .

Eigenvectors and Eigenvalues: An Extension

An eigenvector is a nonzero vector whose direction is preserved by a linear transformation, although its length may change and its direction may reverse. If $𝐯$ is an eigenvector of $A$ , then

$A 𝐯 = λ 𝐯,$

where $λ$ is the corresponding eigenvalue.

Datei:Eigenvectors.gif

For Grades 11–13, the key conceptual idea is more important than a full theory: eigenvectors identify directions that the transformation treats in a particularly simple way. They are important in dynamical systems, Principal component analysis, computer graphics, differential equations, and many models of repeated change.

For a diagonal matrix

$D = (\begin{matrix} 3 & 0 \\ 0 & \frac{1}{2} \end{matrix}),$

the standard basis vectors are eigenvectors. The vector $𝐞_{1}$ is scaled by $3$ , while $𝐞_{2}$ is scaled by $\frac{1}{2}$ .

Connections and Applications

Computer Graphics

Matrices are widely used to transform points and direction vectors in computer graphics. Rotations, scalings, reflections, and shears can be combined into a single product matrix. This is useful because a long sequence of geometric operations can then be applied efficiently.

Pure translations are affine rather than linear transformations in ordinary two-dimensional coordinates. In computer graphics, they can be included in matrix calculations by using Homogeneous coordinates and an extra coordinate.

Physics and Engineering

Vectors describe quantities such as force, velocity, and electric fields. Matrices can describe changes of coordinates, rotations of reference frames, and coupled systems of equations. Understanding what a matrix does geometrically helps you interpret formulas rather than using them mechanically.

Data and Networks

A matrix can organize data or encode relationships between quantities. Repeated multiplication can model transitions, network processes, or discrete-time systems. In more advanced courses, eigenvectors and eigenvalues help identify long-term patterns and dominant directions.

Solving Linear Systems

A system such as

$\begin{matrix} 2 x + y & = 5, \\ x + y & = 3 \end{matrix}$

can be written as

$(\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 3 \end{matrix}) .$

If the coefficient matrix is invertible, you can write

$𝐱 = A^{- 1} 𝐛 .$

For this example, the solution is $x = 2$ and $y = 1$ . This links the geometry of invertible transformations with the algebra of systems that have unique solutions.

Problem-Solving Strategy

When you face a problem about matrices and transformations, ask yourself these questions in order.

What are the input and output spaces? The matrix dimensions tell you how many components enter and how many leave.

What happens to the basis vectors? The columns reveal the transformation directly.

Is the transformation linear? Check whether the origin stays fixed and whether addition and scalar multiplication are preserved.

Is more than one transformation involved? If so, translate the order of actions into the correct matrix product.

What does the determinant say? It reveals area or volume scaling, orientation in the plane, and whether a square matrix is invertible.

Can you check the result geometrically? A quick sketch can expose sign errors, incorrect order, or an impossible result.

Interactive Tasks

Quiz: Test Your Knowledge

What do the columns of a two-by-two transformation matrix represent in the standard basis? (The images of the standard basis vectors) (!The slopes of all transformed lines) (!The determinant and inverse) (!The coordinates of only one input vector)

Which condition must every linear transformation satisfy? (It sends the zero vector to the zero vector) (!It preserves every distance) (!It preserves every angle) (!It moves every point)

Which matrix represents a ninety-degree counterclockwise rotation? (The matrix with rows zero minus one and one zero) (!The identity matrix) (!The matrix with rows one zero and zero minus one) (!The matrix with rows two zero and zero two)

What does the product AB mean when matrices represent transformations? (B acts first and A acts second) (!A and B act at exactly the same time) (!A acts first and B always acts second) (!The order never matters)

What does the absolute value of a two-dimensional determinant measure geometrically? (The factor by which area is scaled) (!The angle of every rotation) (!The length of the first column) (!The number of matrix entries)

What does determinant zero imply for a square transformation matrix? (The transformation is not invertible) (!The transformation is always a rotation) (!The transformation preserves every area) (!The matrix equals the identity)

Which transformation is not linear in ordinary two-dimensional coordinates? (A nonzero translation) (!A rotation about the origin) (!A reflection in a line through the origin) (!A shear)

What is the identity matrix used for? (It leaves every vector unchanged) (!It sends every vector to zero) (!It reflects every vector) (!It exchanges all coordinates)

What is an eigenvector? (A nonzero vector whose direction is preserved up to scaling) (!A vector that must have length one) (!A vector that always maps to zero) (!A row containing the determinant)

Why is matrix multiplication generally not commutative? (Changing transformation order can change the result) (!Every matrix has a different size) (!Matrices cannot represent geometry) (!Only diagonal matrices can be multiplied)

Memory Game

Matrix	Rectangular array that can represent a linear map
Vector	Quantity described by components relative to a basis
Determinant	Signed scale factor for area in two dimensions
Inverse	Transformation that reverses an invertible transformation
Eigenvector	Nonzero direction preserved up to scaling
Basis	Set of independent directions used to describe vectors

Drag and Drop

Match the correct terms.	Topic
Rotation	Preserves lengths and angles while turning vectors about the origin
Scaling	Changes vector components by fixed multiplicative factors
Reflection	Reverses orientation across a line through the origin
Shear	Slants shapes while preserving parallel structure in one direction
Projection	Collapses vectors onto a lower-dimensional subspace

...

Crossword Puzzle

Matrix	What rectangular array can represent a linear transformation?
Vector	What object has components and can be transformed by a matrix?
Rotation	What transformation turns vectors around the origin?
Determinant	What quantity measures signed area scaling?
Inverse	What transformation reverses an invertible transformation?
Eigenvector	What nonzero vector keeps its direction up to scaling?

LearningApps

Cloze Text

A matrix can represent a

between coordinate spaces. The columns of a transformation matrix are the images of the

. A linear transformation must send the zero vector to the

. Matrix multiplication models the

of linear transformations. The order of matrix multiplication matters because it is generally

. The absolute value of a determinant gives the factor by which

changes in two dimensions. A square matrix with determinant zero has no

. An eigenvector keeps its direction up to multiplication by an

.

Open-Ended Tasks

Easy

Transformation Sketch: Choose one two-by-two matrix, transform at least four vectors, and draw the original and transformed vectors on coordinate grids. Explain what pattern you observe.
Basis Vector Investigation: Compute the images of the two standard basis vectors for three different matrices and describe how the columns predict each transformation.
Matrix Story: Write a short explanation for a younger learner that describes a matrix as an action on vectors rather than only as a table of numbers.
Geometry Photo Hunt: Find or photograph examples of rotation, reflection, scaling, or shear in everyday designs and annotate which transformations are approximately represented.

Standard

Transformation Video: Produce a short video or screen recording that demonstrates how a square grid changes under a rotation, scaling, reflection, and shear, including the matrix for each transformation.
Matrix Composition Experiment: Use graphing software or hand calculations to compare two transformations in both possible orders and explain why the final results differ.
Determinant Area Study: Draw several shapes, apply matrices with different determinants, and compare predicted area scale factors with measured or calculated transformed areas.
Linear or Affine Interview: Interview a classmate about whether common geometric moves are linear, record their reasoning, and then create a corrected classification with explanations.

Advanced

Inverse Design Challenge: Design an invertible transformation that sends two chosen independent vectors to two new independent vectors, derive its matrix, find its inverse, and verify both directions.
Eigenvector Exploration: Investigate a two-by-two matrix with two real eigenvectors, visualize several input vectors, and explain what makes the eigenvector directions special.
Graphics Pipeline Project: Model a simple two-dimensional graphic object and create a sequence of matrix transformations that rotates, scales, reflects, or shears it. Combine the transformations into one matrix and compare the results.
Real-World Linear Model: Choose an application from physics, engineering, computing, economics, or data science, research how matrices and linear transformations are used, and present both the mathematics and the limitations of the model.

Learning Assessment

Transformation Analysis: Given an unfamiliar two-by-two matrix, determine the images of basis vectors, sketch the transformation of a square, classify the geometric effect as far as possible, and justify your conclusions.
Composition Reasoning: Compare two products AB and BA for a rotation and a nonuniform scaling, then explain geometrically why the products differ.
Determinant Transfer: Predict what happens to areas and orientation under a matrix with a negative determinant, test your prediction on a triangle, and explain the result.
Invertibility Argument: Decide whether a given square matrix can be reversed without immediately calculating an inverse, using determinant and geometric reasoning.
Model Construction: Construct a matrix that performs a stated transformation on two independent vectors, then verify the matrix on a third vector written as a linear combination of the first two.
Application Critique: Analyze a real or simulated use of a matrix transformation and explain which properties are preserved, which are changed, and what assumptions make the model linear.

Evidence of Learning

Knowledge: You can define matrices, vectors, linear transformations, determinants, inverses, basis vectors, eigenvectors, and composition, and you can connect each term to its geometric meaning.
Procedural skill: You can multiply matrices and vectors, multiply compatible matrices, calculate two-by-two determinants and simple inverses, and construct a transformation matrix from basis images.
Representational skill: You can translate between equations, matrices, vector coordinates, transformed grids, and verbal explanations.
Reasoning: You can explain why matrix multiplication represents composition, why order matters, why determinant zero means information is lost, and why translations are not linear in ordinary coordinates.
Products: Strong evidence may include annotated diagrams, a transformation video, a digital graphing investigation, a written explanation, a research presentation, or a verified matrix model.
Transfer: You can use the same ideas to interpret unfamiliar transformations in geometry, computer graphics, physics, engineering, data, or systems of equations.
Communication: You use mathematical language accurately, show intermediate reasoning, label matrices and vectors clearly, and check whether answers make geometric sense.

OERs on the Topic

For additional open reference material, you can explore the English Wikipedia article on linear maps:

Linked Learning Areas

The topic connects algebraic calculation, geometric visualization, functions, coordinate geometry, systems of equations, computer graphics, and higher linear algebra. The navigation table below summarizes useful next steps.

Matrices and Linear Transformations