Load Packages
Problem Statement
The problem C25, selected page 443,
Define the linear transformation
\(T\quad :\quad { C }^{ 3 }\rightarrow { C }^{ 2 },\quad T(\begin{bmatrix} { x }_{ 1 } \\ { x }_{ 2 } \\ { x }_{ 3 } \end{bmatrix})\quad =\quad \begin{bmatrix} { 2x }_{ 1 }-{ x }_{ 2 }+5{ x }_{ 3 } \\ -4{ x }+2{ x }_{ 2 }-10{ x }_{ 3 } \end{bmatrix}\)
Verify that T is a linear transformation.
Solution
Defination of linear transformation: A linear transformation, \(T:U\rightarrow V\) , is a function that carries elements of the vector space U (called the domain) to the vector space V (called the codomain).
\(T(\begin{bmatrix} { x }_{ 1 } \\ { x }_{ 2 } \\ { x }_{ 3 } \end{bmatrix})\quad =\quad \begin{bmatrix} { 2x }_{ 1 }-{ x }_{ 2 }+5{ x }_{ 3 } \\ -4{ x }+2{ x }_{ 2 }-10{ x }_{ 3 } \end{bmatrix}\\ \quad \quad \quad \quad \quad \quad \quad =\quad { x }_{ 1 }\begin{bmatrix} 2 \\ -4 \end{bmatrix}\quad +\quad { x }_{ 2 }\begin{bmatrix} -1 \\ 2 \end{bmatrix}\quad +\quad { x }_{ 3 }\begin{bmatrix} 5 \\ -10 \end{bmatrix}\\ \quad \quad \quad \quad \quad \quad \quad =\quad \begin{bmatrix} 2 & -1 & 5 \\ -4 & 2 & -10 \end{bmatrix}\begin{bmatrix} { x }_{ 1 } \\ { x }_{ 2 } \\ { x }_{ 3 } \end{bmatrix}\)
We can verify from above that T is a linear transformation.