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Questions and Answers
What defines a finite dimensional vector space?
What defines a finite dimensional vector space?
What is a basis in the context of a finite dimensional vector space?
What is a basis in the context of a finite dimensional vector space?
A nonempty, finite subset that is a spanning set for the vector space and linearly independent.
A finite dimensional vector space has a unique basis.
A finite dimensional vector space has a unique basis.
False
What does the Basis Theorem state?
What does the Basis Theorem state?
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What is stated in the Coordinate Theorem?
What is stated in the Coordinate Theorem?
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What are coordinates in a vector space?
What are coordinates in a vector space?
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In a finite dimensional vector space, a spanning set can have fewer elements than a linearly independent subset.
In a finite dimensional vector space, a spanning set can have fewer elements than a linearly independent subset.
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What does the Dimensionality Theorem imply?
What does the Dimensionality Theorem imply?
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What does the Isomorphism Theorem state?
What does the Isomorphism Theorem state?
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Study Notes
Finite and Infinite Dimensional
- A vector space V is finite dimensional if there exists a nonempty spanning set for V.
- A vector space is infinite dimensional if no nonempty spanning set exists.
Basis
- A basis is a nonempty, finite subset of a finite dimensional vector space V.
- It functions both as a spanning set and is linearly independent.
- A finite dimensional vector space can have infinitely many bases.
Basis Theorem
- Every finite dimensional nonzero vector space must have a basis.
Coordinate Theorem
- For a finite dimensional vector space V with basis {u₁,..., uₙ}, each vector v in V can be expressed as:
- v = k₁u₁ + k₂u₂ + ... + kₙuₙ, where k₁,..., kₙ are unique real numbers in ℝⁿ.
Coordinates and Coordinate Vectors
- The coefficients k₁,..., kₙ represent the coordinates of vector v in the given basis.
- The coordinate vector of v in this basis is denoted as (k₁,..., kₙ) within ℝⁿ.
Rollover Theorem
- In a finite dimensional vector space V, any spanning set with m elements must satisfy:
- m ≥ n, where n is the number of elements in any linearly independent subset of V.
Dimensionality Theorem
- Every basis of a nonzero finite dimensional vector space contains the same number of elements.
- The dimension of the vector space is defined as the number of elements in any of its bases.
Isomorphism Theorem
- Any finite dimensional vector space with dimension n is isomorphic to ℝⁿ, meaning there exists a one-to-one correspondence between them.
Studying That Suits You
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Description
Explore key concepts of finite and infinite dimensional vector spaces and the definition of a basis through this interactive flashcard quiz. Test your understanding of vector space theory and enhance your grasp of linear algebra principles.
