Basis

A rigorous guide to bases, coordinates, change of representation, and proof of spanning plus independence.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

A basis of a vector space $V$ is an ordered list $(v_1,\ldots,v_n)$ that is linearly independent and spans $V$. Every $v\in V$ then has a unique representation $v=c_1v_1+\cdots+c_nv_n$. The coordinate vector is $[v]_B=(c_1,\ldots,c_n)^T$ and depends on the chosen ordered basis $B$.

Notation and mathematical language

The standard basis of $\mathbb R^n$ is $(e_1,\ldots,e_n)$. A change-of-coordinates matrix $P_{C\leftarrow B}$ has columns $[b_j]_C$ and satisfies $[v]_C=P_{C\leftarrow B}[v]_B$. Reordering a basis changes coordinates even though the underlying vectors and space do not change.

Conceptual picture

Spanning ensures enough directions; independence removes redundancy. Uniqueness follows because two coordinate representations subtract to a linear relation among basis vectors, forcing all coefficient differences to zero.

Fully worked example

Interpretation and application

Bases compress a vector into coordinates suited to a problem: Fourier modes, eigenvectors, polynomial coefficients, or geometric axes. Coordinates change with representation, while statements such as linear independence and dimension are intrinsic.