Basis (linear algebra)

File:Basis graph.svg

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 [math]\displaystyle{ \mathbb{R}^3 }[/math] is the vector space then :

B[math]\displaystyle{ = }[/math]{[math]\displaystyle{ (1,0,0),(0,1,0),(0,0,1) }[/math]} is a basis of [math]\displaystyle{ \mathbb{R}^3 }[/math]

It's easy to see that for any element of [math]\displaystyle{ \mathbb{R}^3 }[/math] it can be represented as a combination of the above basis. Let [math]\displaystyle{ x }[/math] be any element of [math]\displaystyle{ \mathbb{R}^3 }[/math], lets say [math]\displaystyle{ x=(x_1,x_2,x_3) }[/math]

Since [math]\displaystyle{ x_1,x_2 }[/math] and [math]\displaystyle{ x_3 }[/math] are elements of [math]\displaystyle{ \mathbb{R} }[/math] then they can be written as [math]\displaystyle{ x_1=1*x_1 }[/math] and so on.

Then the combination equals the element [math]\displaystyle{ x }[/math]

This shows that the set B is a basis of [math]\displaystyle{ \mathbb{R}^3 }[/math]