A basis is the point where generation and uniqueness meet. Dimension is the invariant number of degrees of freedom that survives every change of coordinates.
Bases as coordinate systems
Definition. A finite list \(B=(v_1, \ldots,v_n)\) is a basis of \(V\) if it spans \(V\) and is linearly independent. Equivalently, every \(v\in V\) has a unique coordinate expansion \[ v=a_1v_1+\cdots+a_nv_n. \] The coordinate vector relative to \(B\) is \([v]_B=(a_1, \ldots,a_n)^\top\in F^n\).
Example. The standard basis of \(F^n\) is \(e_1, \ldots,e_n\). The list \((1,x, \ldots,x^m)\) is a basis of \(F_m[x]\). The matrix units \(E_{ij}\) form a basis of \(M_{m,n}(F)\), so \(\dim M_{m,n}(F)=mn\).
Existence of bases
Theorem. In a finitely generated vector space, every spanning list contains a basis and every linearly independent list can be extended to a basis.
Proof. Starting from a spanning list, remove any vector that lies in the span of the others. The span is unchanged, and because the list is finite the process stops at an independent spanning list. Starting from an independent list inside a finite spanning set, add spanning vectors one at a time whenever they are not already in the current span; independence is preserved and the final span is all of \(V\).
The dimension theorem for bases
Theorem. Any two bases of a finite-dimensional vector space have the same number of vectors. This common number is \(\dim V\).
Proof. If \(B\) has \(n\) vectors and \(C\) has \(m\) vectors, then \(B\) is an independent list contained in the span of \(C\), so the exchange lemma gives \(n\le m\). Reversing the roles gives \(m\le n\), hence \(m=n\).
Coordinates and change of basis
Proposition. If \(B=(v_1, \ldots,v_n)\) is a basis, the coordinate map \(\Phi_B:V\to F^n\), \(v\mapsto [v]_B\), is an isomorphism. If \(C\) is another basis, the change-of-basis matrix \(P_{C\leftarrow B}\) is determined by \[ [v]_C=P_{C\leftarrow B}[v]_B. \]
Calculation. To find \(P_{C\leftarrow B}\), put the \(C\)-coordinates of the \(B\)-basis vectors into the columns: \[ P_{C\leftarrow B}=\big[[v_1]_C\ \cdots\ [v_n]_C\big]. \] In \(F^n\) with standard coordinates, if \(S_B\) and \(S_C\) are the matrices whose columns are the basis vectors, then \(P_{C\leftarrow B}=S_C^{-1}S_B\).
Dimension formulas
Proposition. If \(U,W\) are finite-dimensional subspaces of \(V\), then \[ \dim(U+W)=\dim U+\dim W-\dim(U\cap W). \] In particular, \(\dim(U\oplus W)=\dim U+\dim W\).
Proof. Choose a basis of \(U\cap W\), extend it to a basis of \(U\), and separately extend it to a basis of \(W\). The union of the two extended bases spans \(U+W\), and the only overlap has already been counted in \(U\cap W\). Linear independence follows by separating the \(U\) and \(W\) parts.
Remark. In infinite-dimensional spaces, bases still exist with the axiom of choice, but ordinary analysis often uses topological bases, Schauder bases, or orthonormal bases instead. This note mostly uses finite-dimensional bases, where coordinates are unambiguous and algebraic dimension behaves exactly as expected.