# Basis (linear algebra)

subset of a vector space that allows defining coordinates

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. 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.

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

## Example

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

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

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

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

Then the combination equals the element $x$ .

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