Let V be a real inner product space. Under what condition on u, v E V...

Question

Question

Let V be a real inner product space. Under what condition on u, v E V is the following equality valid: where l 1x12 = (x, x) YE V.

math algebra

Answer 1

Answer #1

To find condithos an u,v Ev Hanca v = o it u and vane orthogonal ana condi toi in

Answer 2

Similar Homework Help Questions

Advanced linear algebra, need full proof thanks Let V be an inner product space (real or comple...

advanced linear algebra, need full proof thanks Let V be an inner product space (real or complex, possibly infinite-dimensional). Let {v1, . . . , vn} be an orthonormal set of vectors. 4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R sat...

17] Let V be an n-dimensional real vector space. An inner product on V is a map g : V × V → R satisfying the following propertics The map g is bilinear. That is, for all v, v1, V2, w, w1, W2 CV and all t1,2 ER The map g is symmetric. That is, g(v, w) g(w, v) for all v, weV. The map g is positive definite. That is, g(v,v) 0 for a v e V with equality...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...

Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...

Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space ...

Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product...

Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product on V if and only if T is positive (per our definition of positivity.
b) Let V be a complex vector space, let (,) be an inner product on V,...

b) Let V be a complex vector space, let (,) be an inner product on V, and let 2, y E V be certain vectors. Assume that (x, y) = 2i and (y, y) = 5. Find (< + iy, iy).

Upts) GIve the text of the Spectral Theorem on a real inner product space E (3pts)...

Upts) GIve the text of the Spectral Theorem on a real inner product space E (3pts) Prove that any eigenvalue of a self-adjoint linear map on a complex inner product space is real. 4,) (3pts) Give the definition of a skew-symmetric matrix. X Lexercisebethe car points baseofPandaERaparameter -C )ER . For all = ( 1 ) E R3 and y-(h /2 yE R2 we define the bilinear form ba by 4 y. (3pts) For which value of a, b, is...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator...

Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
Let V be a finite-dimensional inner product space, and let U and W be subspaces of...

Let V be a finite-dimensional inner product space, and let U and W be subspaces of V. Denote dim(V) = n, dim(U) = r, dim(W) = s. Recall that the proj and perp maps with respect to any subspace of V are linear transformations from V to V. Select all statements that are true. Note that not all definitions above may be used in the statements below If proju and perpu are both surjective, then n > 0 If perpw...

Please provide an example where u, v ∈ V (V is an inner product space) s.t....

Please provide an example where u, v ∈ V (V is an inner product space) s.t. ||u+v||^2=||u||^2+||v||^2 but u and v are not orthogonal.