Basis (linear algebra)

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 $V$, any basis of $V$ will have the same number of vectors. This number is called the dimension of $V$.

Example

$B=\{(1,0,0),(0,1,0),(0,0,1)\}$ is a basis of ${ℝ}^3$ as a vector space over $ℝ$.

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

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

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