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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.

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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!
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.
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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.)
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Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim...
Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.
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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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