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Basis (linear algebra)

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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.


If [math]\mathbb{R}^3[/math] is the vector space then:

[math]B=\{(1,0,0),(0,1,0),(0,0,1)\}[/math] is a basis of [math]\mathbb{R}^3[/math].

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

Then the combination equals the element [math]x[/math].

This shows that the set [math]B[/math] is a basis of [math]\mathbb{R}^3[/math].