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).

Expert Solution

Colby Messinger answered 1 year ago

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.

Prove that if U, V and W are vector spaces such that U and V are...

Prove that if U, V and W are vector spaces such that U and V are isomorphic and V and W are isomorphic, then U and W are isomorphic.

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...

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...

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

Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W). Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E...

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

1. Let V and W be vector spaces over R. a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation. b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) =...

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

1. For a map f : V ?? W between vector spaces V and W to be a linear map it must preserve the structure of V . What must one verify to verify whether or not a map is linear? 2. For a map f : V ?? W between vector spaces to be an isomorphism it must be a linear map and also have two further properties. What are those two properties? As well as giving the names...

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.

Let V and W be Banach spaces and suppose T : V → W is a...

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

Question