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Matrix
Representation

Learn how every linear transformation can be captured by a single matrix, turning geometric operations into matrix multiplication.

Linear Algebra

What Is a Linear Transformation?

A linear transformation is a function \( T : \mathbb{R}^n \to \mathbb{R}^m \) that satisfies two key properties for all vectors \( \mathbf{u}, \mathbf{v} \) and scalars \( c \):

Additivity
\( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
Homogeneity
\( T(c\,\mathbf{u}) = c\,T(\mathbf{u}) \)

Visualizing a Transformation

A linear transformation maps the standard grid to a new grid. Lines remain lines, the origin stays fixed, and parallel lines stay parallel.

Input Space \(\mathbf{e}_1\) \(\mathbf{e}_2\) → \(T\) Output Space \(T(\mathbf{e}_1)\) \(T(\mathbf{e}_2)\)

Why Matrices Matter

Instead of describing a transformation with words or formulas, we can encode everything in a compact rectangular array of numbers: a matrix.

Compact
One \(m \times n\) matrix encodes an entire transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\).
Composable
Composing two transformations is just matrix multiplication.
Computable
Computers evaluate matrix-vector products in nanoseconds: \(T(\mathbf{x}) = A\mathbf{x}\).

The Representation Theorem

The matrix \( A \) is called the standard matrix of \( T \). Its columns are the images of the standard basis vectors:

Building the Standard Matrix

\(\mathbf{e}_1\) \(\begin{bmatrix}1\\0\end{bmatrix}\) \(T\) \(T(\mathbf{e}_1)\) \(\begin{bmatrix}a\\c\end{bmatrix}\) \(\mathbf{e}_2\) \(\begin{bmatrix}0\\1\end{bmatrix}\) \(T\) \(T(\mathbf{e}_2)\) \(\begin{bmatrix}b\\d\end{bmatrix}\) → Standard Matrix \(A\) \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) Column 1 of \(A\) Column 2 of \(A\)

Transformation vs. Matrix

Linear Transformation \(T\) Standard Matrix \(A\)
Nature A function between vector spaces A rectangular array of numbers
Definition Given by a rule, e.g. \(T(\mathbf{x})\) Given by entries \(a_{ij}\)
Evaluation Apply the rule to an input Multiply: \(A\mathbf{x}\)
Composition \(S \circ T\) \(BA\) (matrix product)
Uniqueness Many formulas, one function Exactly one standard matrix per \(T\)

How to Find the Standard Matrix

1
Identify the domain and codomain. Determine \(n\) and \(m\) so \(A\) will be \(m \times n\).
2
Write down the standard basis vectors \(\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\) of \(\mathbb{R}^n\).
3
Compute \(T(\mathbf{e}_j)\) for each \(j = 1, 2, \ldots, n\).
4
Assemble the matrix: \(A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{bmatrix}\).
5
Verify by checking \(A\mathbf{x} = T(\mathbf{x})\) for a sample vector \(\mathbf{x}\).

Example 1: Reflection

Find the standard matrix for reflection across the \(x\)-axis: \(T(x_1, x_2) = (x_1, -x_2)\).

1
\(T(\mathbf{e}_1) = T\!\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix}\)
2
\(T(\mathbf{e}_2) = T\!\begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}0\\-1\end{pmatrix}\)
3
Standard matrix: \(A = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\)

Reflection Visualized

\(x_1\) \(x_2\) (3, 2) (3, -2) \(x_1\)-axis

Example 2: Rotation by \(\theta\)

Find the matrix for counterclockwise rotation by angle \(\theta\) in \(\mathbb{R}^2\).

1
\(T(\mathbf{e}_1) = \begin{pmatrix}\cos\theta \\ \sin\theta\end{pmatrix}\): rotate the unit vector along \(x\)-axis.
2
\(T(\mathbf{e}_2) = \begin{pmatrix}-\sin\theta \\ \cos\theta\end{pmatrix}\): rotate the unit vector along \(y\)-axis.
3
Rotation matrix: \(R_\theta = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}\)

Rotation Visualized

\(x_1\) \(x_2\) \(\mathbf{e}_1\) \(T(\mathbf{e}_1)\) \(\mathbf{e}_2\) \(T(\mathbf{e}_2)\) \(\theta\)

Example 3: \(\mathbb{R}^3 \to \mathbb{R}^2\)

Find the standard matrix for \(T(x_1, x_2, x_3) = (x_1 + 3x_3,\; 2x_1 - x_2)\).

1
\(T(\mathbf{e}_1) = T(1,0,0) = \begin{pmatrix}1\\2\end{pmatrix}\)
2
\(T(\mathbf{e}_2) = T(0,1,0) = \begin{pmatrix}0\\-1\end{pmatrix}\)
3
\(T(\mathbf{e}_3) = T(0,0,1) = \begin{pmatrix}3\\0\end{pmatrix}\)

Verifying the Matrix

Let's verify Example 3 with \(\mathbf{x} = \begin{pmatrix}2\\5\\-1\end{pmatrix}\).

Using \(T\) directly
\(T(2, 5, -1) = (2 + 3(-1),\; 2(2) - 5) = (-1, -1)\)
Using \(A\mathbf{x}\)
\(\begin{bmatrix}1&0&3\\2&-1&0\end{bmatrix}\begin{pmatrix}2\\5\\-1\end{pmatrix} = \begin{pmatrix}-1\\-1\end{pmatrix}\)

Common Mistakes

✕ Wrong Dimensions
Writing an \(n \times m\) matrix instead of \(m \times n\). Remember: rows = output, columns = input.
✕ Rows Instead of Columns
Placing \(T(\mathbf{e}_j)\) as rows of \(A\) instead of columns. The images are always columns.
✕ Forgetting Linearity Check
Applying this method to a function that is not linear, such as \(T(\mathbf{x}) = \mathbf{x} + \mathbf{b}\) (affine, not linear).

Summary of Key Transformations

Transformation Standard Matrix Type
Reflection over \(x\)-axis \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) \(2\times2\)
Rotation by \(\theta\) \(\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}\) \(2\times2\)
Projection onto \(x\)-axis \(\begin{bmatrix}1&0\\0&0\end{bmatrix}\) \(2\times2\)
Scaling by \(k\) \(\begin{bmatrix}k&0\\0&k\end{bmatrix}\) \(2\times2\)
Identity \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) \(2\times2\)

Question 1 of 5

What are the columns of the standard matrix for a linear transformation \(T\)?

Question 2 of 5

If \(T : \mathbb{R}^3 \to \mathbb{R}^4\) is linear, what size is its standard matrix?

Question 3 of 5

The standard matrix for the transformation \(T(x_1, x_2) = (3x_1 - x_2,\; x_1 + 4x_2)\) is:

Question 4 of 5

Which of the following is not a linear transformation?

Question 5 of 5

If \(A\) is the standard matrix for \(T\) and \(B\) is the standard matrix for \(S\), what is the standard matrix for the composition \(S \circ T\)?