Question: (a) In \(M_{22}\), the real vector space of \(2 \times 2\) matrices with real entries, find the coordinate vector of \(A=\) \(\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)\) with respect to the basis

\(\mathscr{M}=\left(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)\right)\)

(b) Find the coordinate vector of \(p(x)=2-x+3 x^{2}\) with respect to the basis \(\mathscr{B}=\left(1,1+x, x^{2}-1\right)\) of \(P_{2}\), the real vector space of polynomials in \(x\) with real coefficients and degree at most 2 .

(c) Find the coordinate vector of \(p(x)=2-x+3 x^{2}\) with respect to the basis \(\mathscr{C}=\left(1+x, 1-x, x^{2}\right)\) of \(P_{2}\), the real vector space of polynomials in \(x\) with real coefficients and degree at most 2 .

(d) For the bases \(\mathscr{B}\) and \(\mathscr{C}\) in parts (b) and (c), find the transition matrix \(P_{\mathscr{E} \rightarrow \mathscr{B}}\) so that

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document