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Prob 3. Let T E L(V). Show that (v, u)T :=(Tu, u〉 is an inner product...
Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that j-1 for some nonnegative numbers a,, j-1,.,k, that sum up to 1
Prob 2. Let T be a normal operator on a complex finite-dimensional inner product space V whose distinct eigenvalues are λι, 'Ak E C. For any u E V such that llul-1, show that...
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 real inner product space. Under what condition on u, v E V is the following equality valid: where l 1x12 = (x, x) YE V.
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
Q6 5 Points Let (V, (,)) be an inner product space and T :V + V and S: V + V be self adjoint linear transformations. Show that To S:V + V is self adjoint if and only if S T =To S.
4.11 Let )s F 2n F2n F be defined as (u, v), (u', v)s u .v -vu where u, v, u', v' e F and is the Euclidean inner product on F. Show that )s is an inner product on F. (Note: this inner product is called the symplectic inner product. It is useful in the construction of quantum error-correcting codes.)
4.11 Let )s F 2n F2n F be defined as (u, v), (u', v)s u .v -vu where u,...
Let v and w be vectors in an inner product space V. Show that v is orthogonal to w if and only if ||v + w|| = ||v – w||.
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
6.2.3 Let U be a complex vector space with a positive definite scalar product and S, T e L(U) self-adjoint and commutative, so T-T o S. (i) Prove the identity 11(S iT)(u)ll-llS(11 )11 2 + llT(11)112, 11 e U. (6.2.10) (ii) Show that S ± iT is invertible if either S or T is so. However, the converse is not true. (This is an extended version of Exercise 4.3.4.)
6.2.3 Let U be a complex vector space with a positive...
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...