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

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Colby Messinger answered 2 years ago

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.

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V...

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is finite-dimensional. 1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the corresponding result for matrices in terms of rank 2) If dim(range Y) = dim V, what can you conclude of Y? Give some explanation 3) If dim(range Y) = dim W, what can you conclude of Y? Give some explanation

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.

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.

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....

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 vector space and let U and W be subspaces of V ....

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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).

If V = U ⊕ U⟂ and V = W ⊕ W⟂, and if S1: U...

If V = U ⊕ U⟂ and V = W ⊕ W⟂, and if S1: U → W and S2: U⟂ → W⟂ are isometries, then the linear operator defined for u1 ∈ U and u2 ∈ U⟂ by the formula S(u1 + u2) = S1u1 + S2u2 is a well-defined linear isometry. Prove this.

Let W denote the set of English words. For u, v ∈ W, declare u ∼...

Let W denote the set of English words. For u, v ∈ W, declare u ∼ v provided that u, v have the same length and u, v have the same first letter and u, v have the same last letter. a) Prove that ∼ is an equivalence relation. b) List all elements of the equivalence class [a] c) List all elements of [ox] d) List all elements of [are] e) List all elements of [five]. Can you find more...

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