Basis (linear algebra) facts for kids

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 $\mathbb{R}^3$ is the vector space then:

$B=\{(1,0,0),(0,1,0),(0,0,1)\}$ is a basis of $\mathbb{R}^3$ .

It's easy to see that for any element of $\mathbb{R}^3$ it can be represented as a 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 they can be written as $x_1=1*x_1$ and so on.

Then the combination equals the element $x$ .

This shows that the set $B$ is a basis of $\mathbb{R}^3$ .

Images for kids

Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube $[-1, 1] n$ as a function of dimension, n. Boxplots show the second and third quartiles of this data for each n, red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound given by Eq. (1) and green curve shows a refined estimate.

Basis (linear algebra) facts for kids

Example

Images for kids

See also