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.

Example[change | change source]

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.

Its 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, lets say 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  x_1(1,0,0) + x_2(0,1,0) + x_3(0,0,1) equals the element x

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