Basis of a Vector Space

Basis

Vector space is said to be finite-dimensional if $\exists$ a finite number of vectors such that the vector space $(L,+,.)=(E,+,.)$ where the span $L$ is $L(v_1..)$.

A basis $B=\{v_1...\}$ of $(E,+,.)$ is a set of vectors $\in E$ that verify the following properties:

• They are linearly independent
• They span $E$, i.e. $E = L(v_1..)$

Null Vector and Linear Dependence

If one of the vectors is equal to the null vector, then these vectors are linear dependent

Steinitz Lemma

$(E,+,.)$ = Finite-dimensional vector space $L(v_1..)=E$ = E its span Let $w_1...$ be $s$ linearly independent vectors $\in E$ Follows that $s\le n$, the number of a set of linearly independent vectors cannot be higher than the number of vectors spanning the vector space.

Corollary of the Steinitz Lemma

^continuation $B=\{w_1...\}$ be its basis, it follows that $s\le n$. The vectors composing the basis are linearly independent. For the Steinitzs Lemma, it follows immediately that $s\le n$

Dimension of a Basis

Order of a Basis

Number of vectors composing a basis is said order of a basis All the bases of a vector spaces have the same order

Dimension of a Vector Space

The order of a basis is said dimension is indicated with dim $(E,+,.)$ or dim($E$). The dimension dim $(E,+,.)=n$ of a vector space is:

• the maximum number of linearly independent vectors of E
• the minimum number of vectors spanning E

Linear independence and dimension

The vectors span the vectors if and only if they are linearly independent

Grassmann Formula

Reduction and Extension of a basis

Basis Reduction Theorem - If some vectors are removed a basis of $(E,+,.)$ is obtained Basis Extension Theorem - Let $w_1..$ be $s$ linearly independent vectors of the vector space. If $w_1..$ are not already a basis, they can be extended to a basis (by adding other linearly independent vectors)

Unique Representation

If the $n$ vectors are linearly dependent while $n-1$ is linearly independent, there is a unique way to express one vectors as linear combination of others (How you would represent other lines with one another, identify them with lambdas)

Grassmanns Formula

Let $(U,+,.)$ and $(V,+,.)$ be vector subspace of $(E,+,.)$. Then, $dim(U+V)+dim(U\cap V) = dim(U) + dim(V)$