Let \(\mathrm{P}_{4}\), denote the vector space of all polynomials of degree at most equal to four. Let \(\mathrm{V}\) be the subspace of \(\mathrm{P}_{4}\). spanned by $\mathrm{S}=\left\\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right\\}\( where \)\alpha_{1}=\mathrm{t}^{4}+\mathrm{t}^{2}+2 \mathrm{t}+1$ \(\alpha_{2}=t^{4}+t^{2}+2 t+2, \alpha_{3}=2 t^{4}+t^{3}+t+2\) and \(\alpha_{4}=t^{4}+t^{3}-t^{2}\) \(-\mathrm{t}\). Find a basis for \(\mathrm{V}\).

Let \(\mathrm{V}\) be the subspace of \(\mathrm{R}^{4}\) spanned by $\mathrm{S}=\left\\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}, \alpha_{5}\right\\}\( where \)\alpha_{1}=[1,2,1,2], \alpha_{2}=[2,1,2,1]$ \(\alpha_{3}=[3,2,3,2], \alpha_{4}=[3,3,3,3], \alpha_{5}=[5,3,5,3]\). Find a basis for \(\mathrm{V}\).

Find the rank of matrix A where: $\begin{array}{rrrrr}\text { (i) } \mathrm{A}= & \mid 1 & 3 & 1 & -2 & -3 \mid \\\ & 11 & 4 & 3 & -1 & -4 \\ & 2 & 3 & -4 & -7 & -3 \\ & 13 & 8 & 1 & -7 & -8\end{array}$ (ii) $\begin{array}{rrr}\mathrm{A}= & \mid \begin{array}{rrr}1 & 2 & -3\end{array} \\ & \mid 2 & 1 & 0 \\ & \mid-2 & -1 & 3 \mid \\ & -1 & 4 & -2\end{array}$ (iii) $\begin{array}{rr}\mathrm{A}= & \mid 1 & 3 \mid \\ & \mid 0 & -2 \mid \\\ & \mid 5 & -1 \\ & \mid-2 & 3\end{array} \mid$

Give an example to illustrate the following theorem: The system of n homogeneous linear equations in \(\mathrm{n}\) unknowns, \(\mathrm{AX}=0\), has a nontrivial solution if and only if rank \(\mathrm{A}<\mathrm{n}\)

Show that the matrix $$ \mathrm{A}=\begin{array}{cll} 0 & 1 & 2 \\ 2 & 3 & 4 \\ 14 & 7 & 10 \end{array} $$ is equivalent to \(\mathrm{D}^{33}{ }_{2}\) where $\mathrm{D}^{\mathrm{m}, \mathrm{n}} \mathrm{r}$ denotes the canonical form under equivalence of A. D \(^{\mathrm{m}, \mathrm{n}}_{\mathrm{r}}\) is the echelon form that has one's along the diagonal and zeros elsewhere, and where all the zero rows are consigned to the depths of the matrix.

Let \(\mathrm{A}\) be the matrix $$ \begin{array}{rrrrrr} \mid 0 & 1 & 3 & -2 & -1 & 2 \\ 10 & 2 & 6 & -4 & -2 & 4 \\ \mid 0 & 1 & 3 & -2 & 1 & 4 \\ \mid 0 & 2 & 6 & 1 & -1 & 0 \end{array} $$ Find the determinant rank of \(\mathrm{A}\)