444. Linear Transformations - JulTob/Mathematics GitHub Wiki

Linear transformations are functions between vector spaces that preserve their structure: they are compatible with the operations of sum of vectors and multiplication of a vector by a scalar.

We define a function $f$ between two sets $A$ and $B$ as a law that associates to each element of A one and only one element of $B$ and denote this law as $f ∶ A ⟶ B$.
Set $A$ is called the function's domain, while set $B$ is called the codomain of the function. We define the image of an element $a ∈ A$, the element $f(a) ∈ B$.
The set of images of all $A$ elements is called the image of f and is denoted by Im(f) or sometimes with f(A).

Not all laws that associate elements of a set with elements of another set are functions.

Let $V$ and $W$ be two vector spaces and let $F ∶ V ⟶ W$ be a function. $F$ is called a linear transformation if:

1. $F (𝐮 + 𝐯) = F (𝐮) + F (𝐯)$ for every $𝐮, 𝐯 ∈ V$ ,
2. $F (λ𝐮) = λ·F(𝐮)$ for every $λ ∈ ℝ$ and for every $𝐮 ∈ V$ .

Let $F ∶ V → W$ be a linear transformation; then $F (𝟎_V ) = 𝟎_W$

$F (𝟎_V ) = F(0𝐯) = 0 F(𝐯) = 𝟎_W$

• Identity Map

  $F ∶ V ⟶ V$
  $𝐯↦𝐯$
  Is linear

• Zero Map

  $F ∶ V ⟶ V$
  $𝐯↦𝟎$
  Is linear