\[ T: C^3 \rightarrow C^2, T \left( \left[ \begin{array}{cccc} x1 \\ x2 \\ x3 \end{array} \right] \right) = \left[ \begin{array}{cccc} 2x1 & -x2 & 5x3 \\ -4x1 & +2x2 & -10x3 \end{array} \right] \]
Verify that T is a linear transformation.
Answer:
Basic concept using example. Let say x1 = 3, x2 = 4 and x3 = 5 then
\[ T \left( \left[ \begin{array}{cccc} 3 \\ 4 \\ 5 \end{array} \right] \right) \rightarrow \left[ \begin{array}{cccc} 2(3) & -(4) & 5(5) \\ -4(3) & +2(4) & -10(5) \end{array} \right] \]
\[ T \left( \left[ \begin{array}{cccc} 3 \\ 4 \\ 5 \end{array} \right] \right) \rightarrow \left[ \begin{array}{cccc} 6 & -4 & 25 \\ -12 & 8 & -50 \end{array} \right] \]
\[ T \left( \left[ \begin{array}{cccc} 3 \\ 4 \\ 5 \end{array} \right] \right) \rightarrow \left[ \begin{array}{cccc} 27 \\ -54 \end{array} \right] \]
\({\rm I\!R}3 \rightarrow {\rm I\!R}2\) transformation
\[T (\overline {a} + \overline{b}) = T (\overline{a}) + T( \overline{b}) \]
\[ T (\overline {a} + \overline{b}) = \left[ \begin{array}{cccc} 2a1 + 2b1 - a2 - b2 + 5a3 + 5b3 \\ -4a1 - 4b1 + 2a2 + 2b2 - 10a3 - 10b3 \end{array} \right] \]
\[ T (\overline {a}) = \left[ \begin{array}{cccc} 2a1 - a2 + 5a3 \\ -4a1 + 2a2 - 10a3 \end{array} \right] \]
\[ T (\overline {b}) = \left[ \begin{array}{cccc} 2b1 - b2 + 5b3 \\ -4b1 + 2b2 - 10b3 \end{array} \right] \]
\[ T (\overline {a}) + T (\overline {b}) = T (\overline {a} + \overline {b}) = \left[ \begin{array}{cccc} 2a1 + 2b1 -a2 - b2 + 5a3 + 5b3 \\ -4b1 -4b1 + 2a2 + 2b2 - 10a3 - 10b3 \end{array} \right] \]
\[ c \overline {a} = \left[ \begin{array}{cccc} ca1 \\ ca2 \\ ca3 \end{array} \right] T(c \overline {a}) = \left[ \begin{array}{cccc} 2ca1 - ca2 + 5ca3 \\ -4ca1 + 2ca2 - 10ca3 \end{array} \right] \]
\[ T(c \overline {a}) = c\left[ \begin{array}{cccc} 2a1 - ca2 + 5ca3 \\ -4a1 + 2a2 - 10a3 \end{array} \right] \]
Resources:
- https://www.youtube.com/watch?v=a9LcIKyuHQo
- https://www.youtube.com/watch?v=pQkIYVuacPI