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 Dimension of a given vector space is the number of elements of the basis.

Example

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

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

It's easy to see that for any element of $\mathbb {R} ^{3}$$\mathbb {R} ^{3}$ it can be represented as a combination of the above basis. Let $x$$x$ be any element of $\mathbb {R} ^{3}$$\mathbb {R} ^{3}$, lets say $x=(x_{1},x_{2},x_{3})$$x=(x_{1},x_{2},x_{3})$

Since $x_{1},x_{2}$$x_{1},x_{2}$ and $x_{3}$$x_{3}$ are elements of $\mathbb {R}$$\mathbb {R}$ then they can be written as $x_{1}=1*x_{1}$$x_{1}=1*x_{1}$ and so on.

Then the combination $x_{1}(1,0,0)+x_{2}(0,1,0)+x_{3}(0,0,1)$$x_{1}(1,0,0)+x_{2}(0,1,0)+x_{3}(0,0,1)$ equals the element $x$$x$

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