Basis (linear algebra)

Subset of a vector space that allows defining coordinates
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:

${\displaystyle B=\{(1,0,0),(0,1,0),(0,0,1)\}}$  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 ${\displaystyle \mathbb {R} ^{3}}$  and let ${\displaystyle x=(x_{1},x_{2},x_{3})}$ .

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 ${\displaystyle B}$  is a basis of ${\displaystyle \mathbb {R} ^{3}}$ .