Problem C25 page 443
Define the linear transformation
\[T:\mathbb{C}^3 \rightarrow \mathbb{C}^2, T \bigg(\begin{vmatrix}x_1\\x_2\\x_3\end{vmatrix}\bigg)=\begin{vmatrix} 2x_1-x_2+5x_3 \\ -4x_1+2x_2-10x_3\end{vmatrix}\]
Verify that T is a linear transformation
Solution:
If T is a linear transformation and a and b are vectors and c is a scaler, then the following should be true
Proving number 1
\[a+b=\begin{vmatrix}a_1+b_1\\a_2+b_2\\a_3+b_3\end{vmatrix}\] \[T(a+b)=\begin{vmatrix}2a_1+2b_1-a_2-b_2+5a_3+5b_3\\-4a_1-4b_1+2a_2+2b_2-10a_3-10b_3\end{vmatrix}\]
\[T(a)=\begin{vmatrix}2a_1-a_2+5a_3\\-4a_1+2a_2-10a_3\end{vmatrix}\] \[T(b)=\begin{vmatrix}2b_1-b_2+5b_3\\-4b_1+2b_2-10b_3\end{vmatrix}\]
\[T(a)+T(b)=T(a+b)=\begin{vmatrix}2a_1+2b_1-a_2-b_2+5a_3+5b_3\\-4a_1-4b_1+2a_2+2b_2-10a_3-10b_3\end{vmatrix}\]
Proving number 2
\[ca=\begin{vmatrix}ca_1\\ca_2\\ca_3\end{vmatrix}\]
\[T(ca)=\begin{vmatrix}2ca_1-ca_2+5ca_3\\-4ca_1+2ca_2-10ca_3\end{vmatrix}\] \[T(ca)=c\begin{vmatrix}2a_1-a_2+5a_3\\-4a_1+2a_2-10a_3\end{vmatrix}=cT(a)\] The above transformations satisfy both point 1 and point 2 and proves that T is a linear transformation.