How do I calculate the transformation matrices?

Linear map $\varphi :\mathbb{R}^3 \mapsto \mathbb{R}^3$ is given through $\varphi (\vec{x})=\varphi (x_1,x_2,x_3)=\begin{pmatrix} x_1+2x_2+x_3\\ -x_1+x_3\end{pmatrix}$ given is also $\vec{v}=(4,-1,3)^T$ and B the base of $\mathbb{R}^3$ and C a base of $\mathbb{R}^2$ with $B=\left \{\begin{pmatrix} -1\\ 0\\ 0\end{pmatrix},\begin{pmatrix} 1\\ 1\\ 0\end{pmatrix},\begin{pmatrix} 1\\ 1\\ 1\end{pmatrix}\right \}, C=\left \{ \begin{pmatrix} 1\\ 0\end{pmatrix},\begin{pmatrix} -1\\ 1\end{pmatrix} \right \}$.

a)Determine the mapping matrix A = $[\varphi] _ {E_2,E_3}$ with respect to the canonical bases $E_3$ and $E_2$.

What I have done:

$\begin{bmatrix}0&2&4\\1&-1&0\end{bmatrix}$. I get this by calculating $A = C^{-1}[\varphi]_{C,B}B$, where $[\varphi]_{C,B} = \begin{bmatrix}1&2&1\\-1&0&1\end{bmatrix}.$

b) Determine the mapping matrix A = $[\varphi] _ {E_2,E_3}$ with respect to the canonical bases $E_3$ and $E_2$.

I’m not 100% sure about this, but this is how I would tackle the solution. First, I have a linear transformation $\varphi(\vec{u}) = A\vec{u} = \vec{v}$ with the transformation matrix given as:

$A = \left( \begin{array}{rrr} 1 & 2 & 1\\ -1 & 0 & 1 \end{array}\right )$

Second, I have two bases, B and C. Let us assume for a moment that $\vec{u_B} = B^{-1}\vec{u}$ and $\vec{v_C} = C^{-1}\vec{v}$ which means that the sought after transformation matrix, $A_{B,C}$, and it’s relation to the equation $A\vec{u}=\vec{v}$, can be written as this:

$A_{B,C} \vec{u_B} = \vec{v_C}$

$A_{B,C} B^{-1}\vec{u} = C^{-1}\vec{v}$

$C A_{B,C} B^{-1}\vec{u} = \vec{v}$

$A = C A_{B,C} B^{-1}$

$A_{B,C} = C^{-1} A B$

c) Determine the transformation matrices $T=\left [ id_{\mathbb{R}^3} \right ]_{B,E_3}$ and $S=\left [ id_{\mathbb{R}^2} \right ]_{C,E_2}$ and calculate the mapping matrix $\left [ \varphi \right ]_{C,B}$ using these coordinate transformations and the mapping matrix from (a).

d) Compute $\left [\vec{v} \right ]_{B}$ and find $\left [ \varphi(\vec{v}) \right ]_{C}$ using the matrix $\left [ \varphi \right ]_{C,B}$.

Can someone help me how should I do c) and d)?

• You should check your work. Expressed in the canonical bases, $\phi(1,0,0)^T=(1,-1)^T$, but that’s not what you get if you multiply the matrix that you computed in part a by $(1,0,0)^T$. Have you perhaps written down the questions incorrectly? Parts a and b as you have them ask for the exact same thing. – amd Jul 4 '17 at 22:48