Section 8 Definitions

A function is a system with one output for every input

$A\vec{x}$ map a vector in $\vec{x}$ in $\mathbb{R}^n$ to a vector in $\mathbb{R}^m$

A = $\begin{bmatrix} 1&3&-1&0\\0&1&0&1 \end{bmatrix}$

(1)
\begin{align} x_1 = \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix}, x_2 = \begin{bmatrix} 0\\1\\0\\0 \end{bmatrix}, x_3 = \begin{bmatrix} 0\\0\\1\\0 \end{bmatrix}, x_4 = \begin{bmatrix} 0\\0\\0\\1 \end{bmatrix}, x_5 = \begin{bmatrix} 1\\2\\-1\\-2 \end{bmatrix}, x_6 = \begin{bmatrix} 1\\0\\1\\1 \end{bmatrix} \end{align}

T: $\mathbb{R}^4 \rightarrow \mathbb{R}^2 \text{ } T(\vec{x}) = A\vec{x}$

Image of $\vec{x}$ is $T(\vec{x}) = A\vec{x}$

(2)
\begin{align} T(\vec{x_1}) = \begin{bmatrix} 1\\0 \end{bmatrix}, T(\vec{x_2}) = \begin{bmatrix} 3\\1 \end{bmatrix}, T(\vec{x_3}) = \begin{bmatrix} -1\\0 \end{bmatrix}, T(\vec{x_4}) = \begin{bmatrix} 0\\1 \end{bmatrix}, T(\vec{x_5}) = \begin{bmatrix} 8\\0 \end{bmatrix}, T(\vec{x_6}) = \begin{bmatrix} 0\\1 \end{bmatrix} \end{align}

Do note that $T(\vec{x_4}) = T(\vec{x_6})$. This means that the function T is not one to one, meaning there can be more than one input for the same output, much as a parabola has the same y value for two different x values.

Linear Transformations

A transformation with domain D is Linear if

  1. $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
  2. $T(c\vec{u}) = cT(\vec{u})$
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