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.
The plural of basis is bases. For any vector space [math]\displaystyle{ V }[/math], any basis of [math]\displaystyle{ V }[/math] will have the same number of vectors. This number is called the dimension of [math]\displaystyle{ V }[/math].
Example
[math]\displaystyle{ B=\{(1,0,0),(0,1,0),(0,0,1)\} }[/math] is a basis of [math]\displaystyle{ \mathbb{R}^3 }[/math] as a vector space over [math]\displaystyle{ \mathbb{R} }[/math].
Any element of [math]\displaystyle{ \mathbb{R}^3 }[/math] can be written as a linear combination of the above basis. Let [math]\displaystyle{ x }[/math] be any element of [math]\displaystyle{ \mathbb{R}^3 }[/math] and let [math]\displaystyle{ x=(x_1,x_2,x_3) }[/math]. Since [math]\displaystyle{ x_1,x_2 }[/math] and [math]\displaystyle{ x_3 }[/math] are elements of [math]\displaystyle{ \mathbb{R} }[/math], then we can write [math]\displaystyle{ x = (x_1, x_2, x_3) = x_1(1,0,0) + x_2(0,1,0) + x_3(0,0,1) }[/math]. So [math]\displaystyle{ x }[/math] can be written as a linear combination of the elements in [math]\displaystyle{ B }[/math].
Also, this process would not be possible for any vector [math]\displaystyle{ x }[/math] if an element was removed from [math]\displaystyle{ B }[/math]. So [math]\displaystyle{ B }[/math] is a basis for [math]\displaystyle{ \mathbb{R}^3 }[/math].
The basis [math]\displaystyle{ B }[/math] is not unique; there are infinitely many bases for [math]\displaystyle{ \mathbb{R}^3 }[/math]. Another example of a basis would be [math]\displaystyle{ \{(1,0,0), (0,1,0), (1,1,1)\} }[/math].
Basis (linear Algebra) Media
The same vector can be represented in two different bases (purple and red arrows).
This picture illustrates the standard basis in R2. The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is linearly dependent upon them.
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