Linear Algebra Flashcards: Dimensions & Basis

Questions and Answers

What defines a finite dimensional vector space?

It cannot have any bases.

It must have infinitely many bases.

There exists a nonempty spanning set for the vector space. (correct)

It has an empty spanning set.

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.

False

What does the Basis Theorem state?

A finite dimensional nonzero vector space has a basis. Signup and view all the answers

What is stated in the Coordinate Theorem?

For a finite dimensional vector space V with basis {u₁,..., uₙ} and any v ∈ V, there exist unique k₁,..., kₙ ∈ ℝⁿ with v = k₁u₁ +...+ kₙuₙ. Signup and view all the answers

What are coordinates in a vector space?

k₁,..., kₙ ∈ ℝ are the coordinates of v in the basis. Signup and view all the answers

In a finite dimensional vector space, a spanning set can have fewer elements than a linearly independent subset.

False Signup and view all the answers

What does the Dimensionality Theorem imply?

Every basis of a nonzero finite dimensional vector space has the same number of elements. Signup and view all the answers

What does the Isomorphism Theorem state?

Any finite dimensional vector space with dimension n is isomorphic to ℝⁿ. Signup and view all the answers

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.

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.