Linear Transformations and Matrices - AP Precalculus

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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