Adjoint of a linear operator (2)

In summary, the conversation discusses the relationship between the image of a linear operator and its kernal, as well as the relationship between the rank of a linear operator and its adjoint. It is proven that the image of the adjoint is equal to the orthogonal complement of the kernal, and that the rank of a linear operator is equal to the rank of its adjoint. Additionally, it is noted that the rank of a linear operator plus the nullity of the operator is equal to the dimension of the vector space.
  • #1
kingwinner
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Q: Suppose V is a finite dimensional inner product space and T:V->V a linear operator.
a) Prove im(T*)=(ker T)^(|)
b) Prove rank(T)=rank(T*)

Note: ^(|) is orthogonal complement

For this question, I don't even know how to start, so it would be nice if someone can give me some hints. Thank you!
 
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  • #2
T* is the linear transformation such that, for all u and v, <Tu,v>= <u,T*v> . ker T= {v, Tv= 0). The "orthogonal complement" of ker T is {u |<u, v>= 0 for all v in ker T}. If w is in the image of T* then w= T*v for some v. Then <u, w>= <u,T*v>= <Tu, v> and if u is in the kernal of T, that is equal to what? That proves that im(T*) is a subset of the orthogonal complement of ker T. Prove the other way similarly. Of course, you should know that the rank(T)+ nullity(T)= dimension of V. (Nullity(T)= dimension of kernal of T and rank(T)= dimension of im(T).
 

FAQ: Adjoint of a linear operator (2)

What is the definition of an adjoint linear operator?

An adjoint linear operator is a mathematical concept that represents the transpose of a linear operator, which is a function that maps vectors from one vector space to another in a linear fashion.

Why is the adjoint linear operator important?

The adjoint linear operator is important because it allows us to study and manipulate linear operators in a more efficient manner. It also plays a crucial role in many areas of mathematics, including functional analysis and quantum mechanics.

How is the adjoint of a linear operator calculated?

The adjoint of a linear operator is calculated by taking the transpose of the matrix representation of the operator and then taking the complex conjugate of each element in the matrix.

What are some properties of the adjoint linear operator?

Some properties of the adjoint linear operator include: it is linear, it preserves inner products, and it satisfies the adjoint property (i.e. the adjoint of the adjoint is the original operator).

How is the adjoint linear operator used in applications?

The adjoint linear operator is used in a variety of applications, including solving differential equations, optimization problems, and in quantum mechanics to calculate observables. It is also used in image processing and signal processing to enhance and analyze data.

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