linear operators Definition and 59 Threads

Let T be a linear operator on a finite dimensional vector space V, over the field F. Suppose TU = I, where U is another linear operator on V, and I is the Identity operator. It can ofcourse be shown that T is invertible and the invese of T is nothing but U itself. What I want to know is an...

Suppose T: X -> Y and S: Y -> Z , X,Y,Z normed spaces , are bounded linear operators. Is there an example where T and S are not the zero operators but SoT (composition) is the zero operator?

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

i need to prove the next statement: let S and T be linear operators on a vector space V, then det(SoT)=det(S)det(T). my way is this: let v belong to V, and {e_i} be a basis of V v=e1u1+...+e_nu_n then T(v)=e1T(u1)+...+enT(un)...

Hi, I try to understand the proof for the uncertainty principle for two Hermitian operators A and B in a Hilbert space. My questions are rather general so you don't need to know the specific proof. The first thing I couldn't get into my head was the definition of uncertainty (\Delta...

I'm not sure where to start with these proofs. Any suggestions getting started would be appreciated. 1. Show that is A,B are linear operators on a complex vector space V, then their product (or composite) C := AB is also a linear operator on V. 2. Prove the following commutator...

1. If I: W-->W is the identity linear operator on W defined by I(w) = w for w in W, prove that the matrix of I repect with to any ordered basis T for W is a nXn I matrix, where dim W= n 2. Let L: W-->W be a linear operator defined by L(w) = bw, where b is a constant. Prove that the...

Let U, V, W be inner product spaces. Suppose that T:U\rightarrow V and S:V\rightarrow W are bounded linear operators. Prove that the composition S \circ T:U\rightarrow W is bounded with \|S\circ T\| \leq \|S\|\|T\|