14. What type of transformation occurs when the vector $\langle 1,1\rangle$ is transformed using $\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$ ?
$180^{\circ}$ rotation
no change
reflection over $y$-axis
reflection over $x$-axis
Answer
\(\boxed{\text{The type of transformation that occurs when the vector } \langle 1,1\rangle \text{ is transformed using } \left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right] \text{ is a } 180^{\circ} \text{ rotation.}}\)
Step 1 ：The given matrix is a 2x2 matrix with -1 on the diagonal and 0 elsewhere. This is a special type of matrix that when multiplied with a vector, it negates the vector. In terms of geometric transformation, this corresponds to a 180 degree rotation.
Step 2 ：To confirm this, we can perform the matrix-vector multiplication and observe the result.
Step 3 ：The result of the matrix-vector multiplication is \([-1, -1]\), which is the vector \(\langle 1,1 \rangle\) rotated by 180 degrees. This confirms our initial thought that the given matrix represents a 180 degree rotation.
Step 4 ：\(\boxed{\text{The type of transformation that occurs when the vector } \langle 1,1\rangle \text{ is transformed using } \left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right] \text{ is a } 180^{\circ} \text{ rotation.}}\)
