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.

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milcah answered 1 year ago

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}...

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

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Let Vand W be vector spaces over F, and let B( V, W) be the set...

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Prove the following: Let V and W be vector spaces of equal (finite) dimension, and let...

Prove the following: Let V and W be vector spaces of equal (finite) dimension, and let T: V → W be linear. Then the following are equivalent. (a) T is one-to-one. (b) T is onto. (c) Rank(T) = dim(V).

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V . b. Define similar matrices c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3. d. Show that...

1. Let V and W be vector spaces over R. a) Show that if T: V...

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

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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.

1. For a map f : V ?? W between vector spaces V and W to...

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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).

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