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Questions and Answers
What does Corollary 1 of the Replacement Theorem state?
What does Corollary 1 of the Replacement Theorem state?
- All finite generating sets for V contain n vectors
- Every linearly independent subset of V is a basis
- Every linearly independent subset can be extended to a basis
- Any basis for V contains the same number of vectors (correct)
What is the condition for a generating set to be a basis for vector space V according to Corollary 2?
What is the condition for a generating set to be a basis for vector space V according to Corollary 2?
It must contain exactly n vectors.
Any linearly independent subset of V that contains exactly n vectors is a basis for V.
Any linearly independent subset of V that contains exactly n vectors is a basis for V.
True (A)
What does Corollary 2 say about extending linearly independent subsets?
What does Corollary 2 say about extending linearly independent subsets?
The unique number of vectors in each basis for V is called the ____.
The unique number of vectors in each basis for V is called the ____.
Flashcards
Corollary 1 of Replacement Theorem
Corollary 1 of Replacement Theorem
Any basis of a vector space V contains the same number of vectors.
Corollary 2: Generating set
Corollary 2: Generating set
A generating set for a vector space V is a basis if and only if it contains exactly n vectors.
Linearly independent set (n vectors)
Linearly independent set (n vectors)
Any linearly independent set containing exactly n vectors is a basis for V.
Extending linearly independent subsets
Extending linearly independent subsets
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Dimension of V
Dimension of V
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Study Notes
Replacement Theorem Corollaries
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Basis Consistency: In any vector space V with a finite basis, all bases have the same number of vectors, indicating a fixed dimension.
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Generating Sets and Basis: For a vector space V of dimension n, any finite generating set must have at least n vectors, and if it has exactly n vectors, it qualifies as a basis.
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Linear Independence and Basis: An independent subset of V containing exactly n vectors forms a basis for V, confirming that linear independence is crucial for basing.
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Extension of Independent Sets: Every linearly independent subset of V can be expanded to a full basis of V, demonstrating that independence can form a foundation for the space.
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Vector Space Dimension: The dimension of vector space V is defined as the unique count of vectors found in any basis, establishing a measure for the space's size and complexity.
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