Basis of a vector space
High school
Consider a space of vectors, such as the plane, in which vectors can be added and stretched. A basis of the space is a set of vectors from which every vector in the space can be built in exactly one way, by stretching each of them and adding the results.
Undergraduate
Consider a vector space $V$. A basis of $V$ is a set of vectors in $V$ that is linearly independent and spans $V$.
Examples
Examples
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The vectors $(1, 0)$ and $(1, 1)$ in $\reals^2$
High school
A basis need not be perpendicular: $(x, y) = (x - y)(1, 0) + y (1, 1)$.
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The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$
High school
The vectors $(1, 0)$ and $(0, 1)$ form a basis of $\reals^2$, because every vector $(x, y)$ is $x (1, 0) + y (0, 1)$, in exactly one way. This basis is the standard basis.
Non-examples
Things that may be mistaken for examples.
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The vectors $(1, 0)$ and $(2, 0)$ in $\reals^2$
High school
Both point along the $x$-axis, so no combination of them gives $(0, 1)$.
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The vectors $(1, 0)$, $(0, 1)$, and $(1, 1)$ in $\reals^2$
High school
The vectors $(1, 0)$, $(0, 1)$, and $(1, 1)$ are not a basis of $\reals^2$. They span $\reals^2$, but they are not independent: $(1, 1) = (1, 0) + (0, 1)$, so $(1, 1)$ is redundant.
Properties
- Every vector $v$ in $V$ can be written in exactly one way as a linear combination $v = c_1 b_1 + \dots + c_n b_n$ of the basis vectors. The numbers $c_1, \dots, c_n$ form the coordinate vector of $v$.
- Every vector space has a basis (assuming the axiom of choice), and all bases of the same space have the same number of vectors: its dimension.
- A vector space has many bases. The matrix that represents a linear map depends on which bases are chosen.
Notes
In infinite-dimensional spaces, such as spaces of functions, a basis in this sense (a Hamel basis) allows only finite linear combinations. A Schauder basis also allows infinite sums, which requires a notion of convergence.