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Created page with "==Linear Transformations== <blockquote style="background: white; border: 1px solid black; padding: 1em;"> :'''Definition:''' A transformation <math>T: \mathbb{R}^n \to \mathb..."

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==Linear Transformations==
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Definition:''' A transformation <math>T: \mathbb{R}^n \to \mathbb{R}^m</math> (or operator if <math>T: \mathbb{R}^n \to \mathbb{R}^n</math>) is defined to be <em>linear</em> if the image <math>(w_1, w_2, ..., w_m)</math> is comprised of only linear equations for every mapping <math>(x_1, x_2, ..., x_n)</math>, that is <math>T(x_1, x_2, ..., x_n) = (w_1, w_2, ..., w_m)</math>. For any vectors <math>\vec{u}, \vec{v} \in \mathbb{R}^n</math> and any scalar <math>k</math> a transformation is linear if <math>T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})</math> and <math>T(k\vec{u}) = kT(\vec{u})</math>.
</blockquote>

Let's first look at an example of a linear transformation. Consider the following linear transformation <math>T: \mathbb{R}^2 \to \mathbb{R}^3</math> defined by the following equations:

<div style="text-align: center;"> <math>\begin{align} w_1 = x_1 + 3x_2 \\ w_2 = 2x_1 - x_2 \\ w_3 = -x_1 + 4x_2 \end{align}</math></div>

We note that the equations forming the image, that is <math>w_1</math>, <math>w_2</math>, and <math>w_3</math> are all linear, so this transformation is also considered linear and that <math>T(x_1, x_2) = (x_1 + 3x_2, 2x_1 - x_2, -x_1 + 4x_2)</math>.
For example, if we take the vector <math>\vec{x} = (1, 2)</math> and apply our linear transformation, we obtain a resultant vector <math>\vec{w} = (7, 0, 7)</math>, and we say that <math>(7, 0, 7)</math> is the image of <math>(1, 2)</math> under the linear transformation <math>T</math>.
In general, a linear transformation <math>T: \mathbb{R}^n \to \mathbb{R}^m</math> is generally defined by the following equations:

<div style="text-align: center;"> <math>\begin{align} \\ w_1 = a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ w_2 = a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \quad \vdots\quad \quad \vdots\quad \quad \vdots\quad \quad \quad \quad \vdots \quad\\ w_m = a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \\ \end{align}</math></div>

In matrix notation we can represent this transformation as <math>w = Ax</math>. <math>A</math> is called the <em>standard matrix</em> for the linear transformation <math>T</math>, though sometimes we use the notation <math>[ T ]</math> instead. Either way, the standard matrix is created from the coefficients from the system of linear equations defining the image of <math>T</math>.

<div style="text-align: center;"> <math>\begin{align} \quad \begin{bmatrix} w_1\\ w_2\\ \vdots\\ w_m \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots\\ x_n \end{bmatrix} \end{align}</math> </div>

This linear transformation <math>T</math> is defined by the standard matrix <math>A</math>, so we say that <math>T</math> is multiplication by <math>A</math> and often denote it with the notation <math>T_A (x) = Ax</math>.

Either way, these transformations will geometrically transform some vector or point in <math>\mathbb{R}^n</math> to some other vector or point in <math>\mathbb{R}^m</math>.

==Properties of Linear Transformations==
We've already stated the following two properties in the definition of a linear transformation, but now we will prove their existence.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Property 1:''' If <math>T: \mathbb{R}^n \to \mathbb{R}^m</math> is a linear transformation, then for any vectors <math>\vec{u}, \vec{v} \in \mathbb{R}^n</math> it follows that <math>T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})</math>.
</blockquote>

*'''Proof:''' Suppose that <math>T</math> is a linear transformation and is multiplication by <math>A</math>. Thus it follows that:

<div style="text-align: center;"><math>\begin{align} T(\vec{u} + \vec{v}) = A(u + v) \\ T(\vec{u} + \vec{v}) = Au + Av \\ T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v}) \\ \blacksquare \end{align}</math></div>

<blockquote style="background: white; border: 1px solid black; padding: 1em;">
:'''Property 2:''' If <math>T: \mathbb{R}^n \to \mathbb{R}^m</math> is a linear transformation, then for any vector <math>\vec{u} \in \mathbb{R}^n</math> and any scalar <math>k</math> it follows that <math>T(k\vec{u}) + kT(\vec{u})</math>.
</blockquote>

*'''Proof:''' Suppose that <math>T</math> is a linear transformation and is multiplication by <math>A</math>. Thus it follows that:

<div style="text-align: center;"><math>\begin{align} T(k\vec{u}) = A(ku) \\ T(k\vec{u}) = k(Au) \\ T(k\vec{u}) = kT(\vec{u}) \\ \blacksquare \end{align}</math></div>

==Licensing==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/linear-transformations Linear transformations, mathonline.wikidot.com] under a CC BY-SA license

Linear Transformations - Revision history

Khanh: Created page with "==Linear Transformations== :'''Definition:''' A transformation T: \mathbb{R}^n \to \mathb..."

Khanh: Created page with "==Linear Transformations==
:'''Definition:''' A transformation $T: \mathbb{R}^n \to \mathb..."$