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The matrix representation is $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. In this transformation, the standard basis vector $\hat{i}$ remains fixed at $[1, 0]^T$, while $\hat{j}$ slides diagonally to $[1, 1]^T$.

Conditions: Horizontal shear keeping $\hat{i}$ fixed.; $\hat{j}$ slides diagonally to $[1, 1]^T$.

The matrix representation is $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. In this transformation, the standard basis vector $\hat{i}$ remains fixed at $[1, 0]^T$, while $\hat{j}$ slides diagonally to $[1, 1]^T$.

Conditions: Horizontal shear keeping $\hat{i}$ fixed.; $\hat{j}$ slides diagonally to $[1, 1]^T$.

The matrix representation is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. This is derived by observing where the standard basis vectors land: $\hat{i}$ rotates to $[0, 1]^T$ and $\hat{j}$ rotates to $[-1, 0]^T$.

Conditions: Rotation is 90 degrees counterclockwise.; Working in a 2D plane with standard basis vectors.

The matrix representation is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. This is derived by observing where the standard basis vectors land: $\hat{i}$ rotates to $[0, 1]^T$ and $\hat{j}$ rotates to $[-1, 0]^T$.

Conditions: Rotation is 90 degrees counterclockwise.; Working in a 2D plane with standard basis vectors.