Linear Algebra - Linear Function (Weighted sum)

Definition

f is a linear function if she is defined by <math>f (x) = M * x</math> where:

• M is an R x C matrix
• and <math>f : \mathbb{F}^C \mapsto \mathbb{F}^R</math>

A Linear function can be expressed as a matrix-vector product:

• If a function can be expressed as a matrix-vector product <math>x \mapsto M * x</math> , it has these properties.
• If the function from <math>\mathbb{F}^C</math> to <math>\mathbb{F}^R</math> has these properties, it can be expressed as a matrix-vector product.

Let <math>V</math> and <math>W</math> be vector spaces over a field <math>\mathbb{F}</math> . A function <math>f : V \mapsto W</math> is a linear function if it satisfies two properties:

• Property L1: For every vector v in V and every scalar <math>\alpha</math> in <math>\mathbb{F}</math>

<MATH> f (\alpha.v) = \alpha f(v) </MATH>

• Property L2: For every two vectors u and v in V,

<MATH> f (u + v) = f (u) + f (v) </MATH>

A linear function maps zero vector to zero vector:

• Lemma: If <math>f : U \mapsto V</math> is a linear function then f maps the zero vector of U to the zero vector of V.

The image of a linear function <math>f : V \rightarrow W</math> is a vector space

When <math>f : V \mapsto W</math> is linear <MATH> \begin{eqnarray} f(\alpha_1.v_1 + \dots + \alpha_n.v_n) & = & \alpha_1.f(v_1) + \dots + \alpha_n.f(v_n) \\ & = & \alpha_1.f(w_1) + \dots + \alpha_n.f(w_n) \end{eqnarray} </MATH>

Kernel

Kernel of a linear function f is <math>{v : f (v) = 0}</math>

For a matrix function <math>f (x) = M * x, \text{Ker f} = \text{Null M}</math> where Null M is the null space

Kernel-Image Theorem: For any linear function <math>f : V \mapsto W</math> : <MATH> \href{dimension}{dim} \text{ Ker f } + \href{dimension}{dim } \text{ } \href{#image}{lm} f = \href{dimension}{dim} \text{ V} </MATH>

where:

• dim is dimension
• lm is Linear Algebra - Function (Set)
• ker is the kernel