Discussion week4

The following two conditions proves the definition of linear transformation.

\[\mathbf{T(u + v)} = T(( a + bx + cx^2) + (d + ex + fx^2)) \]

\[ = T((a+d) + (b+e)x + (c+f)x^2)) \]

\[ = \begin{bmatrix}2(a+d) - (b+e)\\(b+e)+(c+f)\end{bmatrix}\]

\[ = \begin{bmatrix}2(a-b) + (2d-e) \\(b+c) + (e+f)\end{bmatrix}\]

\[ = \begin{bmatrix}2(a-b)\\b+c\end{bmatrix} + \begin{bmatrix}2d-e\\e+f\end{bmatrix}\] \[ = T(u) + T(v) \]

\[T(\alpha u) = T (\alpha( a + bx + cx^2))\]

\[ = T((\alpha a) + (\alpha b)x + (\alpha c)x^2)\]

\[ = \begin{bmatrix}2(\alpha a) - (\alpha b) \\ (\alpha b) + (\alpha c)\end{bmatrix}\]

\[ = \begin{bmatrix}\alpha(2a-b) \\ \alpha(b+c)\end{bmatrix}\]

\[ = \alpha \begin {bmatrix}(2a-b) \\ (b+c)\end{bmatrix}\]

\[ = \alpha T (u) \]