Let V be a vector space of dimension 1 over a field k and choose a...

Let V be a vector space of dimension 1 over a field k and choose a fixed nonzero element voe V, which is therefore a basis. Let W be any vector space over k and let woe W be an arbitrary vector. Show that there is a unique linear transformation T: V → W such that T(v)= wo. [Hint: What must T(Avo) be?)

Expert Solution

Colby Messinger answered 2 years ago

(3) Let V be a vector space over a field F. Suppose that a ? F,...

(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0. (4) Prove that for any field F, F is a vector space over F. (5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...

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Let V be a finite-dimensional vector space over C and T in L(V). Prove that the...

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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 ?

Let V and W be finite dimensional vector spaces over a field F with dimF(V )...

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Question 1. Let V and W be finite dimensional vector spaces over a field F with...

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1. Hint 1. Suppose A = {~v1, . . . , ~vn}...

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If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal...

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal basis {v1,v2.....vk-1}. Work out a orthonormal basis of W in terms of that of V and the orthogonal complement of V in W. Provide detailed reasoning.

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