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Questions and Answers
What is a basis 'B' for a vector space V?
What is a basis 'B' for a vector space V?
What must be true for a subset 'B' to be a basis for a vector space V?
What must be true for a subset 'B' to be a basis for a vector space V?
Each element of V can be uniquely expressed in terms of the vectors in B.
If a vector space V is generated by a finite set S, then some subset of S is a basis for V.
If a vector space V is generated by a finite set S, then some subset of S is a basis for V.
True
What does the Replacement Theorem state?
What does the Replacement Theorem state?
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What is a finite-dimensional vector space?
What is a finite-dimensional vector space?
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Match the theorems to their corresponding statements:
Match the theorems to their corresponding statements:
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Every basis for a finite-dimensional vector space contains the same number of vectors.
Every basis for a finite-dimensional vector space contains the same number of vectors.
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What does Corollary 2 state regarding a vector space with dimension n?
What does Corollary 2 state regarding a vector space with dimension n?
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What can be said about the dimension of W and V if dim(W) = dim(V)?
What can be said about the dimension of W and V if dim(W) = dim(V)?
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Any basis for a subspace W can be extended to a basis for the entire space V.
Any basis for a subspace W can be extended to a basis for the entire space V.
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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.
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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.
