Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Solutions

Expert Solution

Colby Messinger answered 10 months ago

Next > < Previous

Related Solutions

Let L be a linear map between linear spaces U and V, such that L: U...

Let L be a linear map between linear spaces U and V, such that L: U -> V and let l_{ij} be the matrix associated with L w.r.t bases {u_i} and {v_i}. Show l_{ij} changes w.r.t a change of bases (i.e u_i -> u'_i and v_j -> v'_j)

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.

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...

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...

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

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

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) =...

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).

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates 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.

Subjects