Linear space

$Linear space$

V

is a Linear Space on

K

:

commutativity:

\forall x, y \in V (x + y = y + x)

associativity:

\forall x, y, z \in V ((x + y) + z = x + (y + z))

\forall x \in V, \forall α, β \in K ((α β) x = α (β x))

additive identity:

\exists0 \in V, \forall v \in V (v + 0 = v)

0 is called the additive identity of V

additive inverse:

\forall x \in V, \exists \overset{x}{ˉ} \in V (x + \overset{x}{ˉ} = 0)

\overset{x}{ˉ} is called the additive inverse of x \in V

multiplicative identity:

1 x = x

distributive properties:

\forall x, y \in V, \forall α, β \in K

α (x + y) = αx + α y and (α + β) x = αx + β x

(a_{1}, ... a_{n}) + (b_{1}, ... b_{n}) = (a_{1} + b_{1}, ..., a_{n} + b_{n})

c \cdot (a_{1}, ... a_{n}) = (c \cdot a_{1}, ..., c \cdot a_{n})

0 := (0, 0, ..., 0)

Let

V

be Linear Space on

K

, then

1. The additive identity of V is unique 2. The additive inverse of x \in V is unique 3.\forall v \in V (0 v = 0 and - 1 v = - v) 4.\forall α \in K (α 0 = 0)

Let

V

the linear space on

K

and

W \subset V

.

W

is a sub linear space of

V

on

K

:

1.0 \in W 2.\forall x, y \in W (x + y \in W) 3. α \in K, v \in V \Rightarrow αv \in W

Sub linear space

W

is a vector space

Suppose

U_{1}, \dots U_{m}

are subsets of

V

U_{1} + \dots + U_{m} := {u_{1} + \dots + u_{m} : u_{1} \in U_{1}, \dots, u_{m} \in U_{m}}

Suppose

U_{1}, \dots, U_{m}

are sub linear spaces of

V

.

U_{1} + \dots + U_{m}

is the smallest sub linear space of

V

containing

U_{1}, \dots, U_{m}

.

Suppose

U_{1}, \dots, U_{m}

are subspaces of

V

.

U_{1} + \dots + U_{m}

is a direct sum:

u \in U_{1} + \dots + U_{m}

can be written in only one way as a sum

u_{1} + \dots + u_{m}

, where each

u_{j}

is in

U_{j}

.

U_{1} \oplus \dots \oplus U_{m} denotes the direct sum of U_{1}, \dots, U_{m} .

Suppose

U_{1}, \dots, U_{m}

are subspaces of

V

. Then

U_{1} + \dots + U_{m}

is a direct sum if and only if the only way to write 0 as a sum

u_{1} + \dots + u_{m}

, where each

u_{j}

is in

U_{j}

, is by taking each

u_{j}

equal to 0 .

Suppose

U

and

W

are subspaces of

V

. Then

U + W

is a direct sum if and only if

U \cap W = {0}

.

W 1 \cap W 2

is a sub linear space of

V

Numerical vector space on

K

: (K^{n}, +, \cdot, 0)

e_{i} = (0, \dots, 1, \dots, 0) (1 ≦ i ≦ n)

canonical basis

: {e_{i} ∣1 ≦ i ≦ n}