Give 2 different examples of an infinite dimensional vector space and provide an explanation in possible....

Give 2 different examples of an infinite dimensional vector space and provide an explanation in possible.

This is a review question for linear algebra and I am trying to better understand the concept.

Thank you!

Expert Solution

Colby Messinger answered 2 years ago

What is a vector space? Provide an example of a finite-dimensional vectors space and an infinite-...

What is a vector space? Provide an example of a finite-dimensional vectors space and an infinite- dimensional vector space.

1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S...

1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S be in- vertible. Prove λ ∈ C is an eigenvalue of T if and only if λ is an eigenvalue of STS−1. Give a description of the set of eigenvectors of STS−1 associated to an eigenvalue λ in terms of the eigenvectors of T associated to λ. Show that there exist square matrices A, B that have the same eigenvalues, but aren’t similar. Hint:...

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1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....

1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?). Show the following are equivalent: (a) The matrix for ? is upper triangular. (b) ?(??) ∈ Span(?1,...,??) for all ?. (c) Span(?1,...,??) is ?-invariant for all ?. please also explain for (a)->(b) why are all the c coefficients 0 for all i>k? and why T(vk) in the span of (v1,.....,vk)? i need help understanding this.

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an...

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?

suppose that T : V → V is a linear map on a finite-dimensional vector space...

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

Give examples–a formula and an illustration–of two-dimensional vector fields F⃗(x,y) with each of the following properties....

Give examples–a formula and an illustration–of two-dimensional vector fields F⃗(x,y) with each of the following properties. You could do the illustrations by hand. a) The direction of F⃗ is constant but the magnitude is not constant. b) The magnitude |F⃗| is constant but the direction is not constant. c) All the vectors F⃗ along a horizontal line are equal, but F⃗ is not constant overall. d) F⃗ (x, y) is perpendicular to xˆi + yˆj at every point (x, y)....

When is it necessary to use a vacuum distillation? Give some explanation and provide examples

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).

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