C26† Verify that the function below is a linear transformation.

\(T:{ P }_{ 2 }\rightarrow { C }^{ 2 },\quad T(a+bx+c{ x }^{ 2 })\quad =\quad \left[ \begin{matrix} 2a-b \\ b+c \end{matrix} \right]\)

The function must adhere to the two properties of a linear transformation: Distributivity across Vector Addition (DVA) and Scalar Multiplication Associativity (SMA).

Distributivity across Vector Addition:

Take two vectors in \({ P }_{ 2 }\):

\(\\ u\quad =\quad [a+bx+c{ x }^{ 2 }]\quad ,\quad v\quad =\quad [e+dx+f{ x }^{ 2 }]\)

Substitute:

\(T(u+v)\quad =\quad T(u)\quad +\quad T(v)\\ =\quad T(a+bx+c{ x }^{ 2 })\quad +\quad (e+dx+f{ x }^{ 2 })\\ =\quad T((a+d)\quad +\quad x(b+e)\quad +\quad { x }^{ 2 }(e+f))\)

Apply function to a, b, c terms:

\(=\quad \left[ \begin{matrix} 2(a+d)\quad - & (b+e) \\ (b+c)\quad + & (e+f) \end{matrix} \right]\)

\(=\quad \left[ \begin{matrix} 2a-b \\ b+c \end{matrix} \right] \quad +\quad \left[ \begin{matrix} 2d-e \\ e+f \end{matrix} \right]\)

The equality shows the first condition is met.

Scalar Multiplication Associativity

Using the scalar \(\alpha\):

\(=\quad T(\alpha u)\)

Substitute:

\(T(\alpha (a+bx+c{ x }^{ 2 }))\\ =T(\alpha a+\alpha bx+\alpha c{ x }^{ 2 }))\)

Apply function to a, b, c terms:

\(=\quad \left[ \begin{matrix} 2\alpha a\quad - & \alpha b \\ \alpha b\quad + & \alpha c \end{matrix} \right]\)

\(=\quad \alpha \left[ \begin{matrix} 2a-b \\ b+c \end{matrix} \right] \quad\)

\(=\quad \alpha T(u)\)

The equality shows the second condition is met.