Linear Algebra Replacement Theorem Corollaries

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Questions and Answers

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?

It must contain exactly n vectors.

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?

<p>Every linearly independent subset can be extended to a basis for V.</p> Signup and view all the answers

The unique number of vectors in each basis for V is called the ____.

<p>dimension of V</p> Signup and view all the answers

Flashcards

Corollary 1 of Replacement Theorem

Any basis of a vector space V contains the same number of vectors.

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)

Any linearly independent set containing exactly n vectors is a basis for V.

Extending linearly independent subsets

Every linearly independent subset of vectors in V can be extended to form a basis.

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Dimension of V

The unique number of vectors in any basis of a vector space V.

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Study Notes

Replacement Theorem Corollaries

Basis Consistency: In any vector space V with a finite basis, all bases have the same number of vectors, indicating a fixed dimension.
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.
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.
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.
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.