Linear Algebra - Linear Function (Weighted sum)

Definition

f is a linear function if she is defined by $f (x) = M * x$ where:

• M is an R x C matrix
• and $f : \mathbb{F}^C \mapsto \mathbb{F}^R$

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

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

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

• Property L1: For every vector v in V and every scalar $\alpha$ in $\mathbb{F}$

$$f (\alpha.v) = \alpha f(v)$$

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

$$f (u + v) = f (u) + f (v)$$

A linear function maps zero vector to zero vector:

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

The image of a linear function $f : V \rightarrow W$ is a vector space

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

Kernel

Kernel of a linear function f is ${v : f (v) = 0}$

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

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

where: