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 ${\displaystyle V}$, any basis of ${\displaystyle V}$ will have the same number of vectors. This number is called the dimension of ${\displaystyle V}$.

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

Any element of ${\displaystyle \mathbb {R} ^{3}}$ can be written as a linear combination of the above basis. Let ${\displaystyle x}$ be any element of ${\displaystyle \mathbb {R} ^{3}}$ and let ${\displaystyle x=(x_{1},x_{2},x_{3})}$. Since ${\displaystyle x_{1},x_{2}}$ and ${\displaystyle x_{3}}$ are elements of ${\displaystyle \mathbb {R} }$, then we can write ${\displaystyle x=(x_{1},x_{2},x_{3})=x_{1}(1,0,0)+x_{2}(0,1,0)+x_{3}(0,0,1)}$. So ${\displaystyle x}$ can be written as a linear combination of the elements in ${\displaystyle B}$.

Also, this process would not be possible for any vector ${\displaystyle x}$ if an element was removed from ${\displaystyle B}$. So ${\displaystyle B}$ is a basis for ${\displaystyle \mathbb {R} ^{3}}$.

The basis ${\displaystyle B}$ is not unique; there are infinitely many bases for ${\displaystyle \mathbb {R} ^{3}}$. Another example of a basis would be ${\displaystyle \{(1,0,0),(0,1,0),(1,1,1)\}}$.

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