[Linear Algebra] Linear Transformation

Transformation

• A transformation, function, or mapping $T$ maps an input $x$ to an output $y$
  • $T : x \mapsto y$
• Domain : set of all the possible values of $x$
• Co-domain : set of all the possible values of $y$
• Image : a mapped output $y$, given $x$
• Range : set of all the output values mapped by each $x$ in the domain

Linear Transformation

• Definition : a transformation (or mapping) $T$ is linear if
  • $T(c \textbf{u} + d \textbf{v}) = cT(\textbf{u}) + dT(\textbf{v})$ for all $\textbf{u}, \textbf{v}$ in the domain of $T$ and for all scalars $c$ and $d$

e.g.

$T : y = 2x + 1$ is not a linear transformation

if $c = 2, d = 3, \textbf{u} = 1, \textbf{v} = 4$

$T(c \textbf{u} + d \textbf{v}) = 2(2 \times 1 + 3 \times 4) + 1 = 29$

$cT(\textbf{u}) + dT(\textbf{v}) = 2\times (2\times1 + 1) + 3\times (2\times4 + 1) = 6 + 27 = 33$

Tips to make a linear transformation

e.g.

$T : y = 2x + 1 = \begin{bmatrix}2 & 1 \\ \end{bmatrix} \begin{bmatrix}x \\ 1 \end{bmatrix}$

if $c = 2, d = 3, \textbf{u} = 1, \textbf{v} = 4$

$T(c \textbf{u} + d \textbf{v}) = \begin{bmatrix}2 & 1 \\ \end{bmatrix} (2 \times \begin{bmatrix}1 \\ 1 \end{bmatrix} + 3 \times \begin{bmatrix}4 \\ 1 \end{bmatrix}) = \begin{bmatrix}2 & 1 \\ \end{bmatrix} \begin{bmatrix}14 \\ 5 \end{bmatrix} = 33$

$cT(\textbf{u}) + dT(\textbf{v}) = 2\times (\begin{bmatrix}2 & 1 \\ \end{bmatrix} \begin{bmatrix}1 \\ 1 \end{bmatrix}) + 3\times (\begin{bmatrix}2 & 1 \\ \end{bmatrix}\begin{bmatrix}4 \\ 1 \end{bmatrix}) = 6 + 27 = 33$