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 plural of basis is bases. For any vector space <math>V</math>, any basis of <math>V</math> will have the same number of vectors. This number is called the dimension of <math>V</math>.

Example

<math>B=\{(1,0,0),(0,1,0),(0,0,1)\}</math> is a basis of <math>\mathbb{R}^3</math> as a vector space over <math>\mathbb{R}</math>.

Any element of <math>\mathbb{R}^3</math> can be written as a linear combination of the above basis. Let <math>x</math> be any element of <math>\mathbb{R}^3</math> and let <math>x=(x_1,x_2,x_3)</math>. Since <math>x_1,x_2</math> and <math>x_3</math> are elements of <math>\mathbb{R}</math>, then we can write <math> x = (x_1, x_2, x_3) = x_1(1,0,0) + x_2(0,1,0) + x_3(0,0,1)</math>. So <math>x</math> can be written as a linear combination of the elements in <math>B</math>.

Also, this process would not be possible for any vector <math>x</math> if an element was removed from <math>B</math>. So <math>B</math> is a basis for <math>\mathbb{R}^3</math>.

The basis <math>B</math> is not unique; there are infinitely many bases for <math>\mathbb{R}^3</math>. Another example of a basis would be <math>\{(1,0,0), (0,1,0), (1,1,1)\}</math>.