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...
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...
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...
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...
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...
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...
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...
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...
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...
(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...
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...
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...
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...
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...
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...
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...
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...
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...
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}...
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...
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...
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...
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...
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.
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...
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...
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...