Basis

$Linear combination$

Let

V

be the linear space on

K

and

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

and

k_{1}, ..., k_{n} \in K

Linear combination of

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

with coefficients

(k_{1}, ..., k_{n})

:

i = 1 \sum n k_{i} v_{i}

$Span$

Span of

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

:

span (v_{1}, \dots, v_{m}) := {i = 1 \sum n k_{i} v_{i} ∣ k_{i} \in K}

$Span is the smallest containing subspace$

The span of a list of vectors in

V

is the smallest subspace of

V

containing all the vectors in the list.

$spans$

v_{1}, \dots, v_{m}

spans

V

:

span (v_{1}, \dots, v_{m}) = V

$Linear independent$

{v_{1}, ..., v_{n}}

is Linear independent :

i = 1 \sum n k_{i} v_{i} = 0 \Rightarrow k_{1} = k_{2} = ... = k_{n} = 0

{v_{1}, ..., v_{n}}

is linear dependent :

{v_{1}, ..., v_{n}} is not linear independent

$Linear Dependence Lemma$

Suppose

v_{1}, \dots, v_{m}

is a linearly dependent list in

V

. Then there exists

j \in {1, 2, \dots, m}

such that the following hold:

(a) v_{j} \in span (v_{1}, \dots, v_{j - 1}) (b) if the j^{th} term is removed from v_{1}, \dots, v_{m}, the span of the remaining list equals span (v_{1}, \dots, v_{m}) .

$Theorem$

1. {v_{1}, ..., v_{n}} is linear independent 2. a_{1} v_{1} + ..., + a_{n} v_{n} = b_{1} v_{1} + ... + b_{n} v_{n} \Rightarrow a_{1} = b_{1}, ... a_{n} = b_{n}

$Finite basis$

{v_{1}, ..., v_{n}}

is finite basis of

V

:

1. {v_{1}, ..., v_{n}} is linear independent 2. {v_{1}, ..., v_{n}} spans V

$Criterion for basis$

A list

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

of vectors in

V

is a basis of

V

if and only if every

v \in V

can be written uniquely in the form

v = a_{1} v_{1} + \dots + a_{n} v_{n}

where

a_{1}, \dots, a_{n} \in F

.

$Spanning list contains a basis$

Every spanning list in a vector space can be reduced to a basis of the vector space.

$Finite linear space$

V

is a finite linear space:

\exists v_{1}, ..., v_{n} \in V (span (v_{1}, ..., v_{n}) = V)

V

is a infinite linear space:

V does not have finite linear space

$Finite-dimensional subspaces$

Every subspace of a finite-dimensional vector space is finite-dimensional.

$Length of linearly independent list and spanning list$

Suppose

u_{1}, \dots, u_{m}

is linearly independent in

V

. Suppose also that

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

spans

V

. Then,

m \leq n

.

$n$ $-dimension$

V

is

n

dimensional linear space:

\exists v_{1}, ..., v_{n} \in V ((v_{1}, ..., v_{n}) is base of V)

$Spanning list contains a basis$

Every spanning list in a vector space can be reduced to a basis of the vector space.

$Basis of finite-dimensional vector space$

Every finite-dimensional vector space has a basis.

$Linearly independent list extends to a basis$

Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space.

$Every subspace of$ $V$ $is part of a direct sum equal to$ $V$

Suppose

V

is finite-dimensional and

U

is a subspace of

V

. Then there is a subspace

W

of

V

such that

V = U \oplus W

.

$Basis length is unique$

V is finite linear space \Rightarrow dim V is uniquely determined

$Dimension$

The dimension of a finite-dimensional vector space is the length of any basis of the vector space.

The dimension of

V

(if

V

is finite-dimensional) is denoted by

dim V

.

$Dimension of a subspace$

If

V

is finite-dimensional and

U

is a subspace of

V

, then

dim U \leq dim V

.

$Spanning list of the right length is a basis$

Suppose

V

is finite-dimensional. Then every spanning list of vectors in

V

with length

dim V

is a basis of

V

.

$Dimension of a sum$

dim (W_{1} + W_{2}) = dim W_{1} + dim W_{2} - dim (W_{1} \cap W_{2})

Linear combination

Span

Span is the smallest containing subspace

spans

Linear independent

Linear Dependence Lemma

Theorem

Finite basis

Criterion for basis

Spanning list contains a basis

Finite linear space

Finite-dimensional subspaces

Length of linearly independent list and spanning list

n-dimension

Spanning list contains a basis

Basis of finite-dimensional vector space

Linearly independent list extends to a basis

Every subspace of V is part of a direct sum equal to V

Basis length is unique

Dimension

Dimension of a subspace

Spanning list of the right length is a basis

Dimension of a sum

$Linear combination$

$Span$

$Span is the smallest containing subspace$

$spans$

$Linear independent$

$Linear Dependence Lemma$

$Theorem$

$Finite basis$

$Criterion for basis$

$Spanning list contains a basis$

$Finite linear space$

$Finite-dimensional subspaces$

$Length of linearly independent list and spanning list$

$n$ $-dimension$

$Spanning list contains a basis$

$Basis of finite-dimensional vector space$

$Linearly independent list extends to a basis$

$Every subspace of$ $V$ $is part of a direct sum equal to$ $V$

$Basis length is unique$

$Dimension$

$Dimension of a subspace$

$Spanning list of the right length is a basis$

$Dimension of a sum$