4. Prove that A = pmatrix 2 & 1 & 0 \ 0 & 3 & 1 \ 0 & 0 & 4 pmatrix and B = pmatrix 2 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 4 pmatrix are similar. (10%)

7. In $\mathbb{R}^4$, let $w_1 = (1, 0, 1, 0)$, $w_2 = (0, 1, 0, 1)$, $w_3 = (1, 0, 0, 0)$. Prove that $\{w_1, w_2, w_3\}$ is linearly independent. Use the Gram-Schmidt process to compute the orthogonal vectors $v_1, v_2$ and $v_3$ such that $span\{w_1, w_2, w_3\} = span\{v_1, v_2, v_3\}$ and then normalize these vectors to obtain an orthonormal set. (15%)

4. Prove that $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}$ are similar. (10%)

相關申論題

相關試卷