Linear transformations and matrices

AP Precalculus· AP Precalculus CED — Functions Involving Parameters, Vectors, and Matrices· 14 min read

1. Definition and Matrix Representation★★☆☆☆⏱ 4 min

📘 Definition

Linear Transformation

$T : R^{2} \to R^{2}$

A function mapping 2D input vectors to 2D output vectors that satisfies two core properties for all vectors $u, v$ and all scalars $c$ : additivity $T (u + v) = T (u) + T (v)$ and homogeneity $T (c v) = c T (v)$

Example:

Any transformation of the form $T (x, y) = (a x + b y, c x + d y)$ is linear

By linearity, any input vector can be written as a combination of the standard basis vectors $[10]$ and $[01]$ , so:

T\left(\begin{bmatrix}x \\ y\end{bmatrix}\right) = xT\left(\begin{bmatrix}1 \\ 0\end{bmatrix}\right) + yT\left(\begin{bmatrix}0 \\ 1\end{bmatrix}\right)

This means outputs of the transformation on the standard basis vectors become the columns of the transformation matrix. For $T (x, y) = (T_{x} (x, y), T_{y} (x, y))$ , the matrix is:

A = \begin{bmatrix} T_x(1,0) & T_x(0,1) \\ T_y(1,0) & T_y(0,1) \end{bmatrix}

📐 Worked Example

Find the matrix representation of the linear transformation $T (x, y) = (4 x - 3 y, 2 x + y)$

1
Evaluate $T$ at the first standard basis vector $[10]$ :
2
$T(1,0) = (4(1) - 3(0), 2(1) + 0) = (4, 2), which becomes the first column of the matrix$
3
Evaluate $T$ at the second standard basis vector $[01]$ :
4
$T(0,1) = (4(0) - 3(1), 2(0) + 1) = (-3, 1), which becomes the second column of the matrix$
5
Assemble and verify the matrix:
6
$A = \begin{bmatrix}4 & -3 \\ 2 & 1\end{bmatrix}, \quad A\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}4x - 3y \\ 2x + y\end{bmatrix}$

Exam tip:

Always test the zero vector first to rule out non-linear transformations quickly on the exam.

2. Composition and Matrix Multiplication★★★☆☆⏱ 4 min

When applying two linear transformations in sequence (first $T_{1}$ , then $T_{2}$ ), the combined transformation $T_{2} \circ T_{1} = T_{2} (T_{1} (v))$ is also linear. If $T_{1}$ has matrix $A_{1}$ and $T_{2}$ has matrix $A_{2}$ , the matrix for the combined transformation is the product $A_{2} A_{1}$ , from the associative property of matrix multiplication.

For two 2x2 matrices, the product $B A$ is calculated by taking the dot product of each row of $B$ with each column of $A$ :

B = \begin{bmatrix}e & f \\ g & h\end{bmatrix}, A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, \quad BA = \begin{bmatrix}ea + fc & eb + fd \\ ga + hc & gb + hd\end{bmatrix}

📐 Worked Example

Let $T_{1}$ be reflection over the y-axis (matrix $A = [- 1 0 01]$ ) and $T_{2}$ be dilation by a factor of 3 (matrix $B = [3003]$ ). Find the matrix for the transformation that first reflects over the y-axis, then dilates by 3.

1
Confirm the order: first $T_{1}$ , then $T_{2}$ = composition $T_{2} \circ T_{1}$ , so the product is $B A$ , not $A B$ .
2
Calculate entry-by-entry:
3
$(3)(-1) + (0)(0) = -3, \quad (3)(0) + (0)(1) = 0 \\ (0)(-1) + (3)(0) = 0, \quad (0)(0) + (3)(1) = 3$
4
The resulting product matrix is:
5
$BA = \begin{bmatrix}-3 & 0 \\ 0 & 3\end{bmatrix}$
6
Verification with input $[24]$ confirms the result matches sequential transformation.

Exam tip:

Always write your order explicitly before multiplying to eliminate order errors.

3. Inverse Transformations and Determinants★★★☆☆⏱ 4 min

A linear transformation is invertible if it is one-to-one and onto, meaning every output vector maps back to exactly one input vector. For a 2x2 matrix $A = [a c b d]$ , invertibility is determined by the determinant $det (A) = a d - b c$ . If $det (A) \neq = 0$ , $A$ is invertible; if $det (A) = 0$ , the transformation cannot be reversed.

The inverse matrix $A^{- 1}$ represents the inverse transformation that undoes $T$ , satisfying $A A^{- 1} = A^{- 1} A = I$ , where $I = [1001]$ is the identity matrix. The formula for the inverse of a 2x2 matrix is:

A^{-1} = \frac{1}{\det(A)} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}

📐 Worked Example

Find the inverse of the transformation matrix $A = [2314]$ , then verify that $A A^{- 1} = I$ .

1
Calculate the determinant first:
2
$\det(A) = (2)(4) - (1)(3) = 8 - 3 = 5 \neq 0, \text{ so inverse exists}$
3
Apply the inverse formula: swap diagonal entries, flip off-diagonal signs, divide by determinant:
4
$A^{-1} = \frac{1}{5}\begin{bmatrix}4 & -1 \\ -3 & 2\end{bmatrix} = \begin{bmatrix}\frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5}\end{bmatrix}$
5
Verify the product to confirm:
6
$\begin{bmatrix}2 & 1 \\ 3 & 4\end{bmatrix}\begin{bmatrix}\frac{4}{5} & -\frac{1}{5} \\ -\frac{3}{5} & \frac{2}{5}\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = I$

Exam tip:

Memorize standard geometric transformation matrices (rotation, reflection, dilation) to save time on the exam.

4. AP-Style Concept Check★★★☆☆⏱ 2 min

✓ Quick check

Test your understanding of core concepts:

Which of the following transformations is a linear transformation?
- A) $T (x, y) = (3 x, y^{2})$
- B) $T (x, y) = (2 x + 5 y, x - 4 y)$
- C) $T (x, y) = (x + 2, y - 3)$
- D) $T (x, y) = (x y, 3 x)$
Reveal answer
B —
Eliminate C first: $T (0, 0) = (2, - 3) \neq = (0, 0)$ , so it is non-linear. A fails homogeneity, D fails additivity. Only B satisfies both linearity properties.

5. Common Pitfalls

Wrong move:

Swapping the order of matrix multiplication for composition, writing $A_{1} A_{2}$ for a transformation that applies $T_{1}$ first then $T_{2}$ .

Why:

Students confuse function notation: $f \circ g = f (g (x))$ , so the first function applied is the inner function, which ends up on the right in matrix form.

Correct move:

Always label the order explicitly: 'First $T_{1}$ , then $T_{2}$ = $T_{2} (T_{1} (v)) = A_{2} A_{1} v$ , so matrix product is $A_{2} A_{1}$ .'

Wrong move:

Forgetting to divide the inverse matrix by the determinant when calculating $A^{- 1}$ .

Why:

Students remember to swap $a$ and $d$ and flip signs of $b$ and $c$ , but omit the scaling factor of $1/ det (A)$ .

Correct move:

After swapping and changing signs, always write the $1/ (a d - b c)$ factor outside the matrix before simplifying entries.

Wrong move:

Calling translation a linear transformation.

Why:

Translations are affine transformations, which look linear, but they do not satisfy the linearity properties.

Correct move:

Always check $T (0)$ : if $T (0) \neq = 0$ , mark the transformation as non-linear immediately.

Wrong move:

Placing the outputs of the standard basis vectors as rows instead of columns in the transformation matrix.

Why:

Students mix up the standard convention for matrix representation.

Correct move:

Always remember: output of $(1, 0)$ is the first column, output of $(0, 1)$ is the second column. Test with a general vector to confirm if unsure.

Wrong move:

Using the wrong sign for $sin θ$ in the rotation matrix.

Why:

Students mix up the direction of rotation.

Correct move:

The standard matrix is defined for counterclockwise $θ$ ; for clockwise rotation, substitute $- θ$ , which flips the sign of the two $sin$ entries.

6. Quick Reference Cheatsheet

Category	Formula	Notes
Linear Transformation Properties	$T (u + v) = T (u) + T (v), T (c v) = c T (v)$	$T (0) = 0$ is required for linearity
Transformation Matrix Construction	Column 1 = $T [10]$ , Column 2 = $T [01]$	Outputs of basis vectors are columns, not rows
2x2 Matrix Multiplication	$[e g f h] [a c b d] = [e a + f c g a + h c e b + f d g b + h d]$	Dot product of rows of first matrix with columns of second
Composition Order	First $T_{1}$ , then $T_{2}$ : Matrix = $A_{2} A_{1}$	First transformation matrix goes on the right
Dilation by factor $k$	$[k 0 0 k]$	Scaling about the origin
Rotation counterclockwise by $θ$	$[cos θ sin θ - sin θ cos θ]$	For clockwise rotation, substitute $- θ$
2x2 Determinant	$det [a c b d] = a d - b c$	$det = 0$ means matrix is not invertible
2x2 Inverse Matrix	$A^{- 1} = \frac{1}{a d - b c} [d - c - b a]$	Only for 2x2 matrices with non-zero determinant

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2024 · MCQ
Identify linear transformation
2023 · FRQ
Composite transformation matrix

What's Next

This topic is the foundation for the remaining content of AP Precalculus Unit 4, including using matrices to solve systems of linear equations and model multi-variable contextual relationships. Next, you will apply linear transformation and matrix conventions to solve linear systems and model real-world processes with multiple input variables. Without mastering the order rules, matrix construction, and inverse formulas covered here, you will constantly make avoidable sign and order errors when working with matrix systems. This topic also provides foundational knowledge for college-level linear algebra, computer graphics, and data science, where linear transformations are core analytical tools.