Basis and Dimension of Subspaces
A basis provides the minimal, non-redundant set of vectors needed to uniquely span a subspace, and its size defines the dimension.
What Is a Basis?
A basis for a vector subspace is a collection of vectors that is both linearly independent and spans the subspace. You can think of a basis as a minimal coordinate framework with zero redundancy. Removing any vector destroys the span, whereas adding any vector introduces linear dependence.
V = \text{span}(v_1, \dots, v_k) \quad \text{where } \{v_1, \dots, v_k\} \text{ is linearly independent}Unique Coordinates
Because a basis has no redundant directions, every vector in the subspace can be written as a linear combination of the basis vectors in exactly one way. The scalar coefficients in this linear combination form the coordinate vector relative to that basis. Changing the basis alters the numerical coordinates, but the geometric vector remains unchanged.
x = c_1 v_1 + c_2 v_2 + \dots + c_k v_kDefining Dimension
The dimension of a subspace is defined as the number of vectors in any of its bases. Remarkably, while a subspace can have infinitely many different bases, every single basis for that subspace will always contain the exact same number of vectors. A subspace consisting of only the zero vector has a dimension of zero.
\dim(V) = k \iff |\mathcal{B}| = kFlashcard
A basis set for a subspace
Check your understanding
Suppose a subspace S in R^4 is spanned by a set of 4 vectors, but these vectors are linearly dependent. What can you conclude about the dimension of S?