Question

\text { Let } \mathbf{v}:=\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right) \text { be the ordered subset of } \mathbb{R}^{3} \text { where } \mathbf{v}_{1}:=\left(\begin{array}{l}

1 \\

4 \\

6

\end{array}\right), \quad \mathbf{v}_{2}:=\left(\begin{array}{l}

0 \\

2 \\

5

\end{array}\right), \quad \mathbf{v}_{3}:=\left(\begin{array}{l}

0 \\

0 \\

3

\end{array}\right) \text { Show that } v \text { forms a basis of } \mathbb{R}^{3} \text {. } ) Calculate the representation matrix of the identity map of R³ under the basis v for thedomain R³ and the standard basis for the range R³ ) Calculate the representation matrix of the identity map of R³ under the standard basis for the domain R³ and the basis v for the range R³. Calculate the representation matrix of the following R-linear map fc under the basis vfor R3 and the standard basis for R?: f_{C}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} ; \quad \mathbf{x}:=\left(\begin{array}{l}

x_{1} \\

x_{2} \\

x_{3}

\end{array}\right) \mapsto f_{C}(\mathbf{x}):=\left(\begin{array}{ccc}

6 & 5 & 4 \\

3 & 2 & 1

\end{array}\right)\left(\begin{array}{l}

x_{1} \\

x_{2} \\

x_{3}

\end{array}\right)

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7