Precalculus · Unit 4: Functions Involving Parameters, Vectors, and Matrices · 14 min read · Updated 2026-05-11
Linear transformations and matrices — AP Precalculus
AP Precalculus · Unit 4: Functions Involving Parameters, Vectors, and Matrices · 14 min read
1. Definition and Matrix Representation★★☆☆☆⏱ 4 min
By linearity, any input vector can be written as a combination of the standard basis vectors $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and $\begin{bmatrix}0 \\ 1\end{bmatrix}$, so:
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:
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 $T_2 \circ T_1 = T_2(T_1(\vec{v}))$ 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 $BA$ 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}
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 $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, invertibility is determined by the determinant $\det(A) = ad - bc$. 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 $AA^{-1} = A^{-1}A = I$, where $I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ is the identity matrix. The formula for the inverse of a 2x2 matrix is:
Exam tip: Memorize standard geometric transformation matrices (rotation, reflection, dilation) to save time on the exam.
4. AP-Style Concept Check★★★☆☆⏱ 2 min
Common Pitfalls
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.
Why: Students remember to swap $a$ and $d$ and flip signs of $b$ and $c$, but omit the scaling factor of $1/\det(A)$.
Why: Translations are affine transformations, which look linear, but they do not satisfy the linearity properties.
Why: Students mix up the standard convention for matrix representation.