Questions and Answers
What does Corollary 1 of the Replacement Theorem state?
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.
True
What does Corollary 2 say about extending linearly independent subsets?
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The unique number of vectors in each basis for V is called the ____.
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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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Description
Explore the essential corollaries of the Replacement Theorem in the context of vector spaces. This quiz covers key definitions and implications for bases and generating sets in linear algebra. Perfect for students reviewing vector space theory.
