Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT...

Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT isomorphic.

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Colby Messinger answered 2 years ago

Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is finite-dimensional....

Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is finite-dimensional. Let X be a 3 × 5 matrix and Y be a 5 × 2 matrix. What are the largest and smallest possible ranks of X, Y, and XY? Give examples of the matrix to support your answers

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give...

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give an example of a map T from R2 to itself (with the usual inner product) such that〈Tv,v〉= 0 for every map. 1B: Suppose that V is a complex space. Show that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉 1C: Suppose T is a linear operator on a complex space such that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all v).

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

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 ]AB is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

V and W are finite dimensional inner product spaces,T:V→W is a linear map, and∗represents the adjoint....

V and W are finite dimensional inner product spaces,T:V→W is a linear map, and∗represents the adjoint. 1A: Let n be a positive integer, and suppose that T is defined on C^n (with the usual inner product) by T(z1,z2,...,zn) = (0,z1,z2,...,zn−1). Give a formula for T*. 1B: Show that λ is an eigenvalue of T if and only if λ is an eigenvalue of T*.

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}...

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is...

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective. Note: Don’t forget that bijective maps are precisely those that have an inverse!

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.)

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.

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation....

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right...

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

Let V be a finite-dimensional vector space over C and T in L(V). Prove that the set of zeros of the minimal polynomial of T is exactly the same as the set of the eigenvalues of T.

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