Basis

A basis is a linearly independent spanning set of vectors in a vector space. Every vector is a unique finite linear combination of basis vectors. Independence supplies uniqueness, and spanning supplies existence. An ordered basis turns the coefficients into a coordinate vector.

For $\mathcal{B}=(b_1,b_2)=((2,1),(1,2))$, the matrix with these vectors as columns isHence the coordinate vector of $v=(5,4)$ is $[v{]}_{\mathcal{B}}=B^{-1}v=(2,1)$, and $B[v{]}_{\mathcal{B}}=v$ recovers the usual coordinates. The determinant $3$ is nonzero, so this recovery works for every vector in ${\mathbb{R}}^2$. A basis need not have perpendicular or unit-length vectors; an orthonormal basis adds those geometric conveniences when an inner product is available.

The point of changing basis is to simplify the operation being studied. Diagonalization uses an eigenbasis to turn a coupled linear map into independent scalings. In a space of polynomials, the basis $1,x,x^2$ makes differentiation easy to encode, while orthogonal polynomials can make approximation errors easier to measure. Gilbert Strang's MIT lecture on independence, basis and dimension develops this connection between attainable columns, redundant columns and coordinates.^[ref]

The unique scalar coefficients in the finite linear combination expressing a vector in a chosen basis. Spanning gives existence, while linear independence makes the difference of two proposed representations have all zero coefficients. Ordering a finite basis turns these coefficients into a coordinate vector.

Let $V$ have finite dimension $n$ and let $S\subseteq V$ have $n$ vectors. Then spanning, linear independence and being a basis are equivalent. From a spanning set, remove redundant vectors until an independent spanning set remains. A removal would give a basis of fewer than $n$ vectors, contradicting dimension invariance. Conversely, if $S$ is independent and a vector lies outside its span, adjoining it gives $n+1$ independent vectors, again impossible. Thus an independent $S$ spans.

A matrix whose columns are the coordinate vectors of an ordered basis in a fixed reference basis. It is an invertible matrix, and multiplication by it converts coordinates in the column basis to the reference coordinates: $v=B[v{]}_B$. For a real inner product space with orthonormal reference basis, its Gram matrix is $B^{T}B$, and its reciprocal basis has basis matrix $B^{-T}$.

An ordered basis is a basis equipped with an ordering of its basis vectors. The order fixes which position of a coordinate vector corresponds to each basis vector.

For an ordered basis $\mathcal{B}=(b_1,\dots ,b_n)$, the coordinate vector $[v{]}_{\mathcal{B}}=(a_1,\dots ,a_n{)}^T$ records the unique coefficients satisfying $v={\sum }_i{a}_i{b}_i$. Change of basis changes this representation while preserving the underlying vector.

For $b_1=(1,0{)}^T$, $b_2=(1,1{)}^T$, the vector $v=(3,1{)}^T$ has coordinate vector $(2,1{)}^T$, since $2b_1+b_2=v$. These are the same coefficients extracted by the dual basis in its worked example. The vector is independent of the chosen basis; its list of coordinates is not. In an orthonormal eigenbasis, the coordinates additionally separate the action of a self-adjoint operator into independent scalar multiplications.

A coordinate of a vector in a chosen basis is one entry of its coordinate vector. The corresponding coordinate functional in the dual basis extracts it. For an orthonormal basis, it can instead be read as an inner product with the basis vector, as in orthonormal eigenbasis. For a nonorthogonal basis, ordinary dot products need not give the coordinates.

A basis vector is one of the vectors in a specified basis of a vector space. Its coordinate column in that basis has one entry equal to one and all other entries zero. Columns of a matrix of a linear map are the images of domain basis vectors, expressed in the chosen codomain basis.

For a basis $e_1,\dots ,e_n$ with cyclic indexing, the vectors $e_j+e_{j+1}$ have coefficient matrix $I+S$, where $S$ cyclically shifts coordinates. Its determinant is $1-(-1{)}^n$. They therefore form a basis exactly when $n$ is odd over a field of characteristic different from two. For even $n$, alternating coefficients produce a nontrivial linear dependence.

Every linearly independent subset of a finite-dimensional vector space extends to a basis. Starting with its vectors, add a vector outside their current span until the span is the whole space. Each addition preserves linear independence, and the process stops because the dimension is finite. This also lets a linear functional on a vector subspace extend to the ambient space by assigning arbitrary values on the additional basis vectors.

The standard basis of the vector space ${\mathbb{R}}^n$ consists of the vectors $e_i$ whose $i$th coordinate is one and whose other coordinates are zero. Every vector $x$ has the unique expression $x={\sum }_i{x}_i{e}_i$ in this basis.

If a vector space is spanned by $n$ vectors, every linearly independent family in it has at most $n$ members. Consequently, any two finite bases have the same number of vectors.

For polynomials in fixed variables and of a fixed degree, the monomials of that degree form a basis of the corresponding homogeneous component.