AP PreCalculus Flashcards: Matrices as Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
What matrix would be used to perform a linear transformation that rotates a vector 90° counterclockwise about the origin?
Using the rotation matrix with θ = 90°, the matrix is $\begin{pmatrix} \\cos 90^\circ & -\\sin 90^\circ \\ \\sin 90^\circ & \\cos 90^\circ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
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What matrix would be used to perform a linear transformation that rotates a vector 90° counterclockwise about the origin?
Using the rotation matrix with θ = 90°, the matrix is $\begin{pmatrix} \\cos 90^\circ & -\\sin 90^\circ \\ \\sin 90^\circ & \\cos 90^\circ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
What does the absolute value of the determinant of a 2x2 transformation matrix represent geometrically?
The absolute value of the determinant gives the magnitude of the dilation, which is the scale factor by which the area of a region in R² is multiplied under the transformation.
What linear transformation is associated with the matrix $\begin{pmatrix} \\cos \\theta & -\\sin \\theta \\ \\sin \\theta & \\cos \\theta \end{pmatrix}$?
This matrix is associated with a linear transformation that rotates every vector counterclockwise by an angle θ about the origin.
What is the relationship between a linear transformation and a matrix?
Every linear transformation can be associated with a unique matrix, where applying the transformation to a vector is equivalent to multiplying the vector by the matrix.
How is the matrix for a composition of two linear transformations found?
The matrix associated with the composition of two linear transformations is the product of the matrices associated with each individual linear transformation.
If a linear transformation L is represented by the matrix A, such that L(v⃗) = Av⃗, how is its inverse transformation L⁻¹ represented?
The inverse transformation L⁻¹ is represented by the inverse matrix A⁻¹, such that L⁻¹(v⃗) = A⁻¹v⃗.
A transformation matrix has a determinant of -4. If this transformation is applied to a shape with an area of 5, what will be the area of the transformed shape?
The new area is the original area multiplied by the absolute value of the determinant. The resulting area is 5 * |-4| = 20.
What is the inverse of a linear transformation?
The inverse of a linear transformation, if it exists, is a transformation that reverses the effect of the original transformation, mapping output vectors back to their original input vectors.
In the context of functions, what role does a matrix play in a linear transformation?
A matrix acts as the rule for a linear function, taking an input vector and producing an output vector through the process of matrix-vector multiplication.
If linear transformation T₁ is represented by matrix A and T₂ by matrix B, what matrix represents the composition of applying T₁ first, then T₂?
The composition of applying T₁ then T₂ is represented by the matrix product BA, as the first transformation's matrix (A) multiplies the vector on the right.
How is the composition of two linear transformations determined?
The composition of two linear transformations is determined by applying one transformation and then applying the second transformation to the result of the first.