AP Precalculus Flashcards: Linear Transformations And Matrices

What this deck covers

This deck focuses on Linear Transformations And Matrices, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the effect of a transformation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ on a vector?

Answer: Rotates the vector 90 degrees counterclockwise. Standard rotation matrix for 90° counterclockwise.

Flashcard 2: What is the effect of a matrix transformation with determinant -1?

Answer: It reflects and possibly rotates the space. Negative determinant indicates orientation reversal.

Flashcard 3: Identify the matrix product $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.

Answer: $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$. Multiply matrices using row-by-column dot products.

Flashcard 4: What is the image of a linear transformation?

Answer: The set of all output vectors. Range of transformation; all possible output vectors.

Flashcard 5: Find the eigenvalues of matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.

Answer: $3, 1$. Solve $\det(A - \lambda I) = 0$ for characteristic polynomial.

Flashcard 6: Calculate the trace of $\begin{pmatrix} 7 & 0 \\ 0 & 3 \end{pmatrix}$.

Answer:

1. Sum diagonal entries: $7 + 3 = 10$.

Flashcard 7: What is the rank of a $3 \times 3$ identity matrix?

Answer:

1. Identity matrix has full rank equal to its dimension.

Flashcard 8: What is the relationship between determinants and area/volume?

Answer: The absolute value of the determinant gives area/volume scale factor. Determinant measures how transformation scales area/volume.

Flashcard 9: Identify the linear transformation represented by $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.

Answer: A scaling by 2 in $x$-direction and 3 in $y$-direction. Diagonal matrix scales each coordinate independently.

Flashcard 10: What is the row-reduced echelon form of $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$?

Answer: $\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}$. Second row is multiple of first, so it reduces to zero.

Flashcard 11: Define the kernel of a linear transformation.

Answer: Set of vectors mapped to zero vector by the transformation. Also called null space; vectors that map to zero.

Flashcard 12: What is the sum of eigenvalues for a matrix $A$?

Answer: Equal to the trace of matrix $A$. Fundamental property relating eigenvalues to trace.

Flashcard 13: What is the effect of a zero determinant on a transformation?

Answer: The transformation collapses the space to a lower dimension. Zero determinant means transformation is not invertible.

Flashcard 14: State the condition for a linear transformation to be invertible.

Answer: The matrix must be square and have a non-zero determinant. Only square matrices with non-zero determinant are invertible.

Flashcard 15: What is the identity element for matrix addition?

Answer: The zero matrix. Adding zero matrix to any matrix leaves it unchanged.

Flashcard 16: What is the row-reduced echelon form of $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$?

Answer: $\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}$. Second row is multiple of first, so it reduces to zero.

Flashcard 17: Calculate the determinant of a $3 \times 3$ identity matrix.

Answer:

1. Identity matrix always has determinant 1 in any dimension.

Flashcard 18: Calculate the trace of matrix $\begin{pmatrix} 5 & 2 \\ 3 & 8 \end{pmatrix}$.

Answer:

1. Trace is sum of diagonal entries: $5 + 8 = 13$.

Flashcard 19: What is the relationship between determinants and area/volume?

Answer: The absolute value of the determinant gives area/volume scale factor. Determinant measures how transformation scales area/volume.

Flashcard 20: What is the definition of an identity matrix?

Answer: A square matrix $I$ with ones on the diagonal and zeros elsewhere. When multiplied by any matrix, returns that same matrix.

Flashcard 21: Identify the matrix representation of $T: \textbf{R}^2 \to \textbf{R}^2$ given $T(1,0) = (3,2)$, $T(0,1) = (1,4)$.

Answer: $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$. Column 1 is $T(1,0)$ and column 2 is $T(0,1)$.

Flashcard 22: What is the effect of a transformation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ on a vector?

Answer: Rotates the vector 90 degrees counterclockwise. Standard rotation matrix for 90° counterclockwise.

Flashcard 23: Compute the determinant of $\begin{pmatrix} 6 & 1 \\ 4 & -2 \end{pmatrix}$.

Answer: -14. Apply formula: $(6)(-2) - (1)(4) = -12 - 2 = -14$.

Flashcard 24: What is the effect of a zero determinant on a transformation?

Answer: The transformation collapses the space to a lower dimension. Zero determinant means transformation is not invertible.

Flashcard 25: What is the definition of a linear transformation?

Answer: A function $T: \textbf{R}^n \to \textbf{R}^m$ preserving vector addition and scalar multiplication. Must satisfy $T(u+v) = T(u) + T(v)$ and $T(cv) = cT(v)$.

Flashcard 26: Find the inverse of matrix $\begin{pmatrix} 2 & 0 \\ 0 & -3 \end{pmatrix}$.

Answer: $\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & -\frac{1}{3} \end{pmatrix}$. For diagonal matrix, inverse has reciprocal diagonal entries.

Flashcard 27: How many solutions exist for a system represented by a singular matrix?

Answer: Either zero or infinitely many solutions. Singular matrices lead to dependent equations.

Flashcard 28: What is the condition for two matrices to be equal?

Answer: They must have the same dimensions and equal corresponding entries. Matrix equality requires identical dimensions and entries.

Flashcard 29: How many solutions exist for a system represented by a singular matrix?

Answer: Either zero or infinitely many solutions. Singular matrices lead to dependent equations.

Flashcard 30: Identify the linear transformation represented by $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.

Answer: A scaling by 2 in $x$-direction and 3 in $y$-direction. Diagonal matrix scales each coordinate independently.

Flashcard 31: What is the standard form of a matrix for a linear transformation?

Answer: Matrix $A$ such that $T(\textbf{x}) = A\textbf{x}$ for all $\textbf{x}$ in $\textbf{R}^n$. Columns of $A$ are images of standard basis vectors.

Flashcard 32: What is the result of multiplying two matrices $A$ and $B$ if they are both $n \times n$ identity matrices?

Answer: An $n \times n$ identity matrix. Identity times identity equals identity.

Flashcard 33: What is the transpose of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$?

Answer: $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$. Swap rows and columns: $(A^T)_{ij} = A_{ji}$.

Flashcard 34: What is the result of any matrix multiplied by the identity matrix?

Answer: The original matrix. Identity matrix is multiplicative identity for matrices.

Flashcard 35: What is the rank of a $3 \times 3$ identity matrix?

Answer:

1. Identity matrix has full rank equal to its dimension.

Flashcard 36: Compute the determinant of $\begin{pmatrix} 6 & 1 \\ 4 & -2 \end{pmatrix}$.

Answer: -14. Apply formula: $(6)(-2) - (1)(4) = -12 - 2 = -14$.

Flashcard 37: What does it mean if a matrix is singular?

Answer: Its determinant is zero. Singular matrices are not invertible and collapse dimensions.

Flashcard 38: Describe a shear transformation in terms of a matrix.

Answer: A transformation that slants the shape of an object. Shear keeps one direction fixed while slanting the other.

Flashcard 39: What is the definition of a linear transformation?

Answer: A function $T: \textbf{R}^n \to \textbf{R}^m$ preserving vector addition and scalar multiplication. Must satisfy $T(u+v) = T(u) + T(v)$ and $T(cv) = cT(v)$.

Flashcard 40: What does it mean if a matrix is orthogonal?

Answer: Its transpose is its inverse. Orthogonal matrices preserve lengths and angles.

Flashcard 41: Determine if matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is its own inverse.

Answer: Yes, $A^2 = I$. Check if $A \cdot A = I$; matrix swaps coordinates.

Flashcard 42: Calculate the trace of matrix $\begin{pmatrix} 5 & 2 \\ 3 & 8 \end{pmatrix}$.

Answer:

1. Trace is sum of diagonal entries: $5 + 8 = 13$.

Flashcard 43: What is the effect of a matrix transformation with determinant -1?

Answer: It reflects and possibly rotates the space. Negative determinant indicates orientation reversal.

Flashcard 44: State the associative property for matrix multiplication.

Answer: $(AB)C = A(BC)$ for matrices $A$, $B$, and $C$. Matrix multiplication is associative but not commutative.

Flashcard 45: What is the condition for two matrices to be equal?

Answer: They must have the same dimensions and equal corresponding entries. Matrix equality requires identical dimensions and entries.

Flashcard 46: What is the result of multiplying two matrices $A$ and $B$ if they are both $n \times n$ identity matrices?

Answer: An $n \times n$ identity matrix. Identity times identity equals identity.

Flashcard 47: What property does a diagonal matrix have regarding eigenvectors?

Answer: The unit vectors are its eigenvectors. Diagonal entries are eigenvalues with standard basis eigenvectors.

Flashcard 48: Describe a shear transformation in terms of a matrix.

Answer: A transformation that slants the shape of an object. Shear keeps one direction fixed while slanting the other.

Flashcard 49: What property does a diagonal matrix have regarding eigenvectors?

Answer: The unit vectors are its eigenvectors. Diagonal entries are eigenvalues with standard basis eigenvectors.

Flashcard 50: Calculate the determinant of a $3 \times 3$ identity matrix.

Answer:

1. Identity matrix always has determinant 1 in any dimension.

Flashcard 51: What is the sum of eigenvalues for a matrix $A$?

Answer: Equal to the trace of matrix $A$. Fundamental property relating eigenvalues to trace.

Flashcard 52: Find the eigenvalues of matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.

Answer: $3, 1$. Solve $\det(A - \lambda I) = 0$ for characteristic polynomial.

Flashcard 53: Calculate the trace of $\begin{pmatrix} 7 & 0 \\ 0 & 3 \end{pmatrix}$.

Answer:

1. Sum diagonal entries: $7 + 3 = 10$.

Flashcard 54: What is the transpose of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$?

Answer: $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$. Swap rows and columns: $(A^T)_{ij} = A_{ji}$.

Flashcard 55: State the associative property for matrix multiplication.

Answer: $(AB)C = A(BC)$ for matrices $A$, $B$, and $C$. Matrix multiplication is associative but not commutative.

Flashcard 56: What does it mean if a matrix is orthogonal?

Answer: Its transpose is its inverse. Orthogonal matrices preserve lengths and angles.

Flashcard 57: What does it mean if a matrix is singular?

Answer: Its determinant is zero. Singular matrices are not invertible and collapse dimensions.

Flashcard 58: What is the result when a linear transformation is applied twice?

Answer: The square of the transformation matrix applied to a vector. Composition of transformations corresponds to matrix multiplication.

Flashcard 59: What is the result when a linear transformation is applied twice?

Answer: The square of the transformation matrix applied to a vector. Composition of transformations corresponds to matrix multiplication.

Flashcard 60: Identify the matrix product $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.

Answer: $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$. Multiply matrices using row-by-column dot products.

Flashcard 61: Find the inverse of the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Answer: $\begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}$. Use formula $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

Flashcard 62: What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

Answer: $ad - bc$. Formula for $2 \times 2$ matrix determinant.