Linear Algebra: 1.6 - Bases and Dimension

Basis

• A basis 'B' for a vector space V is a linearly independent subset that generates V.
• Vectors in 'B' are essential for forming the entire vector space V.

Theorem 1.8

• A subset B = {u_1, u_2, ..., u_n} of a vector space V serves as a basis if each vector v in V can be uniquely expressed as a linear combination of the basis vectors.
• The uniqueness of scalars a_1, a_2, ..., a_n is crucial to defining a basis.

Theorem 1.9

• If a vector space V is generated by a finite set S, then at least one subset of S is a basis for V, indicating V has a finite basis.

Theorem 1.10 (Replacement Theorem)

• For a vector space generated by a set G with n vectors and a linearly independent subset L with m vectors, the condition m ≤ n holds true.
• There exists a subset H of G with n - m vectors such that L ∪ H generates V, ensuring the structure of the vector space is preserved.

Corollary 1

• Every basis of a finite-dimensional vector space V consists of the same number of vectors, establishing consistency across different bases.

Finite-Dimensional

• A vector space is finite-dimensional if it has a basis with a finite number of vectors.
• The dimension of V, denoted dim(V), is defined by the number of vectors in its basis, whereas spaces without finite bases are labeled infinite-dimensional.

Corollary 2

• For a vector space V with dimension n:
  • Any finite generating set must have at least n vectors; a set with exactly n vectors is a basis.
  • Any linearly independent set with n vectors in V serves as a basis.
  • Any linearly independent subset can be extended to form a basis for V.

Theorem 1.11

• A subspace W of a finite-dimensional vector space V is also finite-dimensional, with dim(W) ≤ dim(V).
• If dim(W) equals dim(V), then W and V are identical.

Corollary

• Any basis for a subspace W of a finite-dimensional vector space V can be extended to a basis for V, preserving the relationship between bases and subspaces.