Linear operator

$Linear map$

f

is a linear map:

additivity:

\forall u_{1} u_{2} \in U, f (u_{1} + u_{2}) = f (u_{1}) + f (u_{2})

homogeneity:

\forall u \in U, \forall α \in K, f (α u) = α f (u)

The set of all linear maps from V to W is denoted L (V, W) .

Suppose

v_{1}, \dots, v_{n}

is a basis of

V

and

w_{1}, \dots, w_{n} \in W

. Then there exists a unique linear map

T : V \to W

such that

T v_{j} = w_{j}

for each

j = 1, \dots, n

.

f

is a linear isomorphism:

1. f is a linear map 2. f is bijective

f

is a Linear transformation:

f : V \to V

is a linear map

1 ker f := {x ∣ x \in U, f (x) = 0} 2 Im f := {y ∣ y \in V, y = f (x)}

(1) Im f \subset V, ker f \subset U (2) f is Surjective \Leftrightarrow Im f = V (3) f is injective \Leftrightarrow ker f = {0}

dim U = dim (Im f) + dim (ker f)

Dual space

V^{*}

:

{f : V \to K ∣ f is linear map}

Dual space is

n

-dimensional linear space on

K