# Basis (linear algebra)

subset of a vector space, such that every vector is uniquely expressible as a linear combination over this set of vectors
This picture illustrates the standard basis in R2. The red and blue vectors are the elements of the basis; the green vector can be given with the basis vectors.

In linear algebra, a basis is a set of vectors in a given vector space with certain properties:

• One can get any vector in the vector space by multiplying each of the basis vectors by different numbers, and then adding them up.
• If any vector is removed from the basis, the property above is no longer satisfied.

The Dimension of a given vector space is the number of elements of the basis.

## Example

If ${\displaystyle \mathbb {R} ^{3}}$  is the vector space then :

B${\displaystyle =}$ {${\displaystyle (1,0,0),(0,1,0),(0,0,1)}$ } is a basis of ${\displaystyle \mathbb {R} ^{3}}$

It's easy to see that for any element of ${\displaystyle \mathbb {R} ^{3}}$  it can be represented as a combination of the above basis. Let ${\displaystyle x}$  be any element of ${\displaystyle \mathbb {R} ^{3}}$ , lets say ${\displaystyle x=(x_{1},x_{2},x_{3})}$

Since ${\displaystyle x_{1},x_{2}}$  and ${\displaystyle x_{3}}$  are elements of ${\displaystyle \mathbb {R} }$  then they can be written as ${\displaystyle x_{1}=1*x_{1}}$  and so on.

Then the combination equals the element ${\displaystyle x}$

This shows that the set B is a basis of ${\displaystyle \mathbb {R} ^{3}}$