What is the Adjoint of a Linear Operator?

In summary, the discussion revolves around a linear operator T on a finite dimensional vector space, and the relationship between its null spaces and range. It is concluded that if the dimension of the range of T is greater than 0, then the null space of T is a subset of the null space of T squared, but this is only true for non-trivial cases.
  • #1
rsa58
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Homework Statement


T is a linear operator on a finite dimensional vector space. then N(T*T)=N(T). the null space are equal.


Homework Equations





The Attempt at a Solution


this is my method, but its does not work if dim(R(T))=0. I'm only concerned with showing
N(T*T) [tex]\subseteq[/tex] N(T). let x beong to N(T*T) then <T*T(x),y>=0=<T(x),T(y)> for all y in the vector space. thus, if dim(R(T)) > 0 then there exists y such that T(y) is not equal to zero so T(x)=0.

any other methods out there?
 
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  • #2
okay i think i got it if dim(R(T))=0 then ofcourse x is in the null space of T.
 
  • #3
But that's only true for the trivial case, the 0 matrix.
 

1. What is the adjoint of a linear operator?

The adjoint of a linear operator is a mathematical concept that represents the transpose of a linear operator in a vector space. It is denoted by a superscript asterisk (A*) and is defined as the linear operator that satisfies the property ⟨Ax, y⟩ = ⟨x, A*y⟩ for all vectors x and y in the vector space.

2. How is the adjoint of a linear operator calculated?

The adjoint of a linear operator can be calculated using the adjoint matrix method, which involves taking the transpose of the matrix representation of the linear operator and then taking the complex conjugate of each element in the matrix. Another method is using the inner product definition, which involves finding the linear operator that satisfies the property ⟨Ax, y⟩ = ⟨x, A*y⟩ for all vectors x and y.

3. What is the significance of the adjoint of a linear operator?

The adjoint of a linear operator has several important applications in mathematics, physics, and engineering. It is used to solve problems related to eigenvalues and eigenvectors, as well as in the study of self-adjoint operators, which have important properties in quantum mechanics and differential equations. It also plays a crucial role in the theory of Hilbert spaces and functional analysis.

4. What is the relationship between the adjoint of a linear operator and its inverse?

The adjoint of a linear operator is closely related to its inverse. If a linear operator has an inverse, then its adjoint is equal to the inverse of its transpose. In other words, (A*)^-1 = (A^-1)^T. This property is useful when solving systems of linear equations using the adjoint matrix method.

5. Can the adjoint of a linear operator have complex eigenvalues?

Yes, the adjoint of a linear operator can have complex eigenvalues. However, for self-adjoint operators, which are symmetric with respect to the inner product, the eigenvalues are always real. In general, the eigenvalues of the adjoint of a linear operator can be complex or real, depending on the properties of the original linear operator.

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