Understanding Basis of a Set in Linear Algebra

A basis for a vector space V is an independent generating set. There are intuitive ways to confirm if a set is a basis, such as checking if it is independent and generates V. The dimension of V helps determine if a subset is a basis. Examples and methods like the extension theorem are explored to find or verify bases of sets in linear algebra.

• Linear Algebra
• Basis Set
• Vector Space
• Dimension

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1. Confirming that a set is a Basis

2. Intuitive Way Definition: A basis B for V is an independent generation set of V. Is Ca basis of V? Independent? yes Generation set? difficult generates V

3. Another way Find a basis for V Given a subspace V, assume that we already know that dim V = k. Suppose S is a subset of V with k vectors If S is independent If S is a generation set S is basis S is basis Is Ca basis of V? Dim V = 2 (parametric representation) C is a subset of V with 2 vectors Cis a basis of V Independent? yes

4. Assume that dim V = k. Suppose S is a subset of V with k vectors Another way If S is independent S is basis By the extension theorem, we can add more vector into S to form a basis. However, S already have k vectors, so it is already a basis. If S is a generation set S is basis By the reduction theorem, we can remove some vector from S to form a basis. However, S already have k vectors, so it is already a basis.

5. Example Is Ba basis of V? Independent set in V? yes Bis a basis of V. Dim V = ? 3

6. Example B is a subset of V with 3 vectors Is Ba basis of V = Span S? 0 0 2/3 1/3 2/3 0 1 0 0 0 0 1 0 0 ??= ???? = 3 1 0 1 0 0 0 0 1 0 0 0 0 1 0 ??????????? ??= Bis a basis of V.