0.1 Practice Problems

1.

Determine whether a transformation defined as \(T(\langle a, b \rangle ) = a + b\) is linear. How about \(T(\langle a, b \rangle ) = ab\)?

2.

For square matrices \(A\) and \(B\) of order \(n \times n\), show that if \(\lambda \) is an eigenvalue of \(AB\), then the eigenvalues of \(BA\) also include \(\lambda \) [MATH02-3].

3.

Find bases for the kernel and the image of \(\begin {bmatrix} 1 & 1 & 1 \\ 0 & 1 & 0 \end {bmatrix}\).

4.

Show that \(A \cong B\) if and only if \(A^\intercal \cong B^\intercal \).

5.

Find the eigenvalues and corresponding eigenspace for \(\begin {bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 1 & 2 \end {bmatrix}\).