Problem 3. An isometry between inner-product spaces V and W is a linear operator L in...

Problem 3. An isometry between inner-product spaces V and W is a linear
operator L in B (V ,W) that preserves norms and inner-products. If x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V .
Suppose that V and W are both real, n-dimensional inner-product spaces.
Thus the scalar field for both is R and both of them have a basis consisting of
n elements. Show that V and W are isometric by demonstrating an isometry
between them.
Hint: take both bases, and cite some linear algebra result that says that
you can orthonormalize them. Prove (or cite someone to convince me) that you
can define a linear function by specifying its action on a basis. Finally, define
your isometry by deciding what it should do on an orthonormal basis for V , and
prove that it preserves inner-products (and thus norms).

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

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).

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*.

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)

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...

4. Verify that the Cartesian product V × W of two vector spaces V and W...

4. Verify that the Cartesian product V × W of two vector spaces V and W over (the same field) F can be endowed with a vector space structure over F, namely, (v, w) + (v ′ , w′ ) := (v + v ′ , w + w ′ ) and c · (v, w) := (cv, cw) for all c ∈ F, v, v′ ∈ V , and w, w′ ∈ W. This “product” vector space (V ×...

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V...

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V , i.e., <T(u),T(v)> = <u,v>, ∀u,v ∈ V , is an isomorphism.

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 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.

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.

Let f : V mapped to W be a continuous function between two topological spaces V...

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V...

Subjects