Linear Algebra: 1.6 - Bases and Dimension

Study Notes

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.

Description

This quiz explores key concepts in bases and dimension from Linear Algebra Chapter 1.6. You'll learn about the definition of a basis for a vector space and related theorems. Perfect for students looking to solidify their understanding of these fundamental topics.