Matrix Representations

The matrix $$\begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix}$$ represents a rotation of 45 degrees about the origin in the $xy$-plane because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the rotation and $$(x, y)$$ are the coordinates of the point before the rotation.

The matrix $$\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$$ represents a reflection about the $x$-axis because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the reflection and $$(x, y)$$ are the coordinates of the point before the reflection.

The matrix $$\begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}$$ represents a shear transformation that maps the line $y = x$ to the line $y = 2x$ because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the shear transformation and $$(x, y)$$ are the coordinates of the point before the shear transformation.

The matrix $$\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}$$ represents a projection onto the $x$-axis because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the projection and $$(x, y)$$ are the coordinates of the point before the projection.

The matrix $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$ represents a change of basis from the standard basis to the basis $${\begin{bmatrix} 1 \ 2 \end{bmatrix}, \begin{bmatrix} 3 \ 4 \end{bmatrix}}$$ because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point in the new basis and $$(x, y)$$ are the coordinates of the point in the old basis.

The matrix $$\begin{bmatrix} -5 & 3 \ 4 & -2 \end{bmatrix}$$ represents the inverse of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ because it satisfies the following equation: $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} \begin{bmatrix} -5 & 3 \ 4 & -2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$$

The matrix $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$ represents the transpose of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ because it is obtained by interchanging the rows and columns of the original matrix.

The determinant of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ is $$-2$$ because it is calculated by subtracting the product of the elements on the main diagonal from the product of the elements on the other diagonal.

The trace of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ is $$7$$ because it is calculated by adding the elements on the main diagonal.

The rank of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{bmatrix}$$ is $$2$$ because it is the maximum number of linearly independent rows or columns in the matrix.

The null space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} -2 \ 1 \end{bmatrix}}$$ because it is the set of all vectors that are orthogonal to the row space of the matrix.

The column space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} 1 \ 3 \end{bmatrix}, \begin{bmatrix} 2 \ 4 \end{bmatrix}}$$ because it is the set of all linear combinations of the columns of the matrix.

The row space of the matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ is $${\begin{bmatrix} 1 \ 2 \end{bmatrix}, \begin{bmatrix} 3 \ 4 \end{bmatrix}}$$ because it is the set of all linear combinations of the rows of the matrix.

The eigenvalues of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ are $$2$$ and $$7$$ because they are the roots of the characteristic polynomial of the matrix.

The eigenvectors of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$ are $$\begin{bmatrix} 1 \ 1 \end{bmatrix}$$ and $$\begin{bmatrix} 1 \ 2 \end{bmatrix}$$ because they are the vectors that are not changed by the matrix except for a scalar multiple.