Basis (linear algebra)

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 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 $\mathbb {R} ^{3}$ as a vector space over $\mathbb {R}$.

Any element of $\mathbb {R} ^{3}$ can be written as a linear 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 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 $\mathbb {R} ^{3}$.

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