# Search results

1. ### Polar decompositions

About the equation T=Ssqrt(T*T), what purpose does the sqrt(T*T) serve?
2. ### Isometries math problem

Homework Statement Prove or give a counterexample: if S ∈ L(V) and there exists an orthonormal basis (e1, . . . , en) of V such that llSejll = 1 for each ej , then S is an isometry. Homework Equations The Attempt at a Solution Can't think of a counterexample. I am assuming that...
3. ### Another positive operator proof

Homework Statement Suppose that T is a positive operator on V. Prove that T is invertible if and only if <Tv,v > is >0 for every v ∈ V \ {0}. Homework Equations The Attempt at a Solution If T is invertible, then TT-1=I.Now let v=v1+...+vn and let Tv=a1v1+...+anvn. Now <Tv...
4. ### Positive integer proof

positive operator proof Homework Statement Prove that if T ∈ L(V) is positive, then so is Tk for every positive integer k. Homework Equations The Attempt at a Solution Let v=b1v1+...+bnvn. Now since T is positive, T has a positive square root. T=S^2. <S^2v, v>=<S^2v1...
5. ### Positive operators

Homework Statement Prove that the sum of any two positive operators on V is positive. Homework Equations The Attempt at a Solution This problem seems pretty simple. But I could be wrong. Should I name two positive operators T and X such that T=SS* and X=AA*? I have a bad history of seeing...
6. ### Making T self adjoint

Suppose U is a finite-dimensional real vector space and T ∈ L(U). Prove that U has a basis consisting of eigenvectors of T if and only if there is an inner product on U that makes T into a self-adjoint operator. The question is, what exactly do they mean by "makes T into a self adjoint...
7. ### Normal operators and self adjointness

Homework Statement Suppose V is a complex inner-product space and T ∈ L(V) is a normal operator such that T9 = T8. Prove that T is self-adjoint and T2 = T. Homework Equations The Attempt at a Solution Consider T9=T8. Now "factor out" T7 on both sides to get T7T2 =TT7. Now we represent T as...
8. ### Normal operators

Homework Statement Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real. Homework Equations The Attempt at a Solution Let c be an eigenvalue. Now since T=T*, we have <TT*v, v>=<v, TT*v> if and only if TT*v=cv on both...
9. ### Another proof on self adjointness

Homework Statement Prove that there does not exist a self-adjoint operator T ∈ L(R3) such that T(1, 2, 3) = (0, 0, 0) and T(2, 5, 7) = (2, 5, 7). Homework Equations The Attempt at a Solution I'm having trouble seeing that there is an actual operator, self adjoint or not, that can do...
10. ### Prove that P is an orthogonal projection if and only if P is self adjoint.

Homework Statement Suppose P ∈ L(V) is such that P2 = P. Prove that P is an orthogonal projection if and only if P is self-adjoint. Homework Equations The Attempt at a Solution Let v be a vector in V. Let w be a vector in W and u be a vector in U and let U and W be subspaces of V where dim...

Homework Statement Show that if V is a real inner-product space, then the set of self-adjoint operators on V is a subspace of L(V). Homework Equations The Attempt at a Solution Let M be the matrix representing T. Since we are dealing with real numbers, and T is self-adjoint, T=T* so M=MT...

Homework Statement Prove or give a counterexample: the product of any two selfadjoint operators on a finite-dimensional inner-product space is self-adjoint. Homework Equations The Attempt at a Solution I'd say that if we let a diagonal matrix represent T (after all, its transpose...
13. ### Conjugate homogeneity

(aT)∗ = \bar{a}T∗ for all a ∈ C and T ∈ L(V,W); This doesn't make much sense to me. Isn't a supposed to be=x+iy and \bar{a}=x-iy? Not a fan of complex numbers. And this proof also confuses me. 7.1 Proposition: Every eigenvalue of a self-adjoint operator is real. Proof: Suppose T is a...
14. ### Proof in linear algebra

Homework Statement Prove that dim null T∗ = dim null T + dimW − dimV and dim range T∗ = dim range T for every T ∈ L(V,W). Homework Equations The Attempt at a Solution I have my solution written down, but just to make sure... I think that nullT*=0 since W is a subspace of V and mapping from...
15. ### Prove that T is injective if and only if T* is surjective

Homework Statement T ∈ L(V,W). Thread title. Homework Equations The Attempt at a Solution Note that T* is the adjoint operator. But there's one thing that I need to get out of the way before I even start the proof. Now consider <Tv, w>=<v, T*w> w in W, v in V. Now when they say T...
16. ### Question about a proof.

Suppose T ∈ L(V) and U is a subspace of V. Prove that U is invariant under T if and only if U⊥ is invariant under T∗. Now for reference, L(V) is the set of transformations that map v (a vector) from V to V. T* is the adjoint operator. The case where the dimension of U is less than V bugs me...
17. ### Proof based on projections

Homework Statement Suppose T ∈ L(V) and U is a subspace of V. Prove that U is invariant under T if and only if PUTPU = TPU. Homework Equations The Attempt at a Solution Consider u\inU. Now let U be invariant under T. Now let PU project v to U so that PU(v)=u. Therefore TPU(v)=T(u). Now...
18. ### Another question about projections.

Say we have a transformation T\inL(V). Now suppose a subspace of V (U) is in the rangespace of T. Now suppose PUv=u with u=a1u1+...+amvm. Now apply T to u to get T(u)=b1u1+...+bmum=/=a1u1+...+amvm. What would happen if we apply PUto T(u)? In other words, what would we end up with after...
19. ### Orthogonal projection

Homework Statement Let P\inL(V). If P^2=P, and llPvll<=llvll, prove that P is an orthogonal projection. Homework Equations The Attempt at a Solution I think that regarding llPvll<=llvll is redundant. For example, consider P^2=P and let v be a vector in V. Doesn't P^2=P kind of give it away...
20. ### L.a. proof

Homework Statement prove that dimV=dimU\bot+dimU Homework Equations The Attempt at a Solution I've done this on paper, and set V=nullT+rangeT where T maps a vector from V to U. Is it safe to assume that nullT and U\bot are the same? Reasoning is that <T(wi), T(uj)>=0 with wi in nullT and...
21. ### Question about orthogonal projections.

Aren't all projections orthogonal projections? What I mean is that lets say there is a vector in 3d space and it gets projected to 2d space. So [1 2 3]--->[1 2 0] Within the null space is [0 0 3], which is perpendicular to every vector in the x-y plane, not to mention the inner product of [0...
22. ### Proof about inner product spaces

Homework Statement Suppose V is a real inner-product space and (v1, . . . , vm) is a linearly independent list of vectors in V. Prove that there exist exactly 2^m orthonormal lists (e1, . . . , em) of vectors in V such that span(v1, . . . , vj) = span(e1, . . . , ej) for all j ∈ {1, . . . , m}...
23. ### Inner product proof

Homework Statement Prove that (\sumajbj)2\leq\sumjaj2*\sum(bj)2/j with j from 1 to n. for all real numbers a1...an and b1...bn Homework Equations The Attempt at a Solution I can prove this using algebra, but how is it done using inner product concepts? If someone could start me up...
24. ### Clueless about inner products

Hello, just started reading about inner products, and they don't make much sense to me (I mean, even basic properties). I read something about the dot product, then they started getting into <x+y, z> is <x,z>+<y,z> what is the purpose of doing this? I'm almost completely clueless about...
25. ### Eigenvector proof

Homework Statement If v and w are eigenvectors with different (nonzero) eigenvalues, prove that they are linearly independent. Homework Equations The Attempt at a Solution Define an operator A such that a is an nxn matrix, and Av=cIv with c an eigenvalue and v and eigenvector. Define a...
26. ### Linear algebra proof (surprise!)

Homework Statement Prove that if eigenvectors v1, v2...vn are such that for any eigenvalue c of A, the subset of all these eigenvectors belonging to c is linearly independent, then the vectors v1,v2..vn are linearly independent. Homework Equations The Attempt at a Solution One...
27. ### Small proof

Homework Statement Show that A and AT share the same eigenvalue. Homework Equations The Attempt at a Solution let v be the eigenvector Av=Icv since ATv=ITcv and IT=I, ATv=Icv so ATv=Icv=Av so A and AT must have the same eigenvalue.
28. ### Linear algebra proof with operator T^2 = cT

Homework Statement Let T:V--->V be an operator satisfying T^2=cT c=/=0. Show that V=U\opluskerT U={u l T(u)=cu} Homework Equations The Attempt at a Solution Now before I start, just one quick question about ker T: U seems to be an eigenspace since T(u)=cu with c the eigenvalue. But that...
29. ### Interesting proof

Homework Statement Let U be a fixed nxn matrix, and consider the operator T:Msub(n,n)---->Msub(n,n) given by T(A)=UA (look familiar?:biggrin:) Show that if dim[Esub(c)(U)]=d then dim[Esub(c)(T)]=nd. Homework Equations The Attempt at a Solution The author provided a small hint. He suggested...
30. ### Eigenvalue proof

Homework Statement Let U be a fixed nxn matrix and consider the operator T: Msub(n,n)------>Msub(n,n) given by T(A)=UA. Show that c is an eigenvalue of T if and only if it is an eigenvalue of U. Homework Equations The Attempt at a Solution If T(A)=UA then T(A)-UA=0 (T-U)A=0. Let...