Define $T : \mathbb R^3 \rightarrow \mathbb R^2~|~T(x,y,z)=(4x+5y+6z, ~~~7x+8y+9z).$ Suppose $\phi_1,\phi_2$ denote the dual basis of the standard basis of $\mathbb R^2$ and $\omega_1,\omega_2,\omega_3$ denote the dual basis of the standard basis of $\mathbb R^3$.
(i) Describe the linear functionals $T'(\phi_1)$ and $T'(\phi_2),~ T'$ represents the dual map of $T$ .
(ii) Write $T'(\phi_1)$ and $T'(\phi_2)$. as a linear combination of $\omega_1,\omega_2,\omega_3$
Attempt: Part(i) We know that $T'(\phi_1)=\phi_1 \circ T$ and $T'(\phi_2)=\phi_2 \circ T$.
If $\{e_1,e_2,e_3\}$ and $\{e_1,e_2\}$ represent the standard basis of $\mathbb R^3$ and $\mathbb R^2$ respectively, Then:
$\omega_1(e_1)=1,~w_1(e_2)=0,~w_1(e_3)=0$
$\omega_2(e_1)=0,~w_2(e_2)=1,~w_2(e_3)=0$
$\omega_3(e_1)=0,~w_3(e_2)=0,~w_3(e_3)=1$
Similarily: $\phi_1(e_1)=1,~\phi_1(e_2)=0$
$\phi_2(e_1)=0, \phi_2(e_2)=1$
Thus, $T'(\phi_1)(x,y,z)= \phi_1 \circ T (x,y,z)= \phi_1 \big (4x+5y+6z, ~~~7x+8y+9z \big )= 4x+5y+6z$
Similarily: $T'(\phi_2)(x,y,z)=7x+8y+9z $
Part(ii) $T'(\phi_1)= c_1 \omega_1+c_2 \omega_2+c_3\omega_3$
Thus, $(T' \circ \phi_1~)(x,y,z)= \big( c_1 \omega_1+c_2 \omega_2+c_3\omega_3 \big ) (x+y+z)=c_1x+c_2y+c_3z$
Did I Attempt this correctly? I am not sure about either of the parts!
Any help will be deeply appreciated! Thanks a lot for reading!