A question in prooving liniar operator

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In summary, the conversation is about a formula for a polynomial and the use of the transpose operator in a 2x2 vector. The purpose of the formula is T(A)= AT- A and there is some confusion about the notation T and the meaning of the little line sign. It is suggested to check with a teacher or consult a book for clarification.
  • #1
transgalactic
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  • #2
I'm not sure what you mean by "for a polynomial". The formula is T(A)= AT- A.

Perhaps you are thinking that AT is a power? Even if it were, that would still be alright- if you can multiply matrices you can certainly take a matrix to a power- and a product of matrices still represents a linear transformation.

However, AT is the standard notation for the 'transpose' of a matrix: basically you swap rows and columns. For a 2 by 2 matrix
[tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T= \left[\begin{array}{cc} a & c \\ b & d\end{array}\right][/tex]
so that
[tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right]= \left[\begin{array}{c c} a & c \\ b & d\end{array}\right]- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right][/itex]
[tex]= \left[\begin{array}{cc} 0 & c- b \\ b- c & 0\end{array}\right][/tex]
 
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  • #3
no this sign is a little line like a derivative sign
no way it can't be a T
??
 
  • #4
then I have no idea what it means- it's not a standard notation. Does your book give a definition? If not, ask your teacher.
 
  • #5
i trust your judgement probably its a print mistake
and they ment T
 
  • #6
It's my vision I'm not sure you should trust! Is this matrix over the real numbers or complex numbers? Sometimes a little "sword" superscript is use to represent the "Hermitian conjugate" where you take the transpose (switch rows to columns) and take the complex conjugate of all entries. Of course, if your matrix has only real number entries, that is the same as the transpose.
 
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1. What is a linear operator?

A linear operator is a mathematical function that maps one vector space to another, preserving the linear structure of the original space. It is used to describe the relationship between two vector spaces and is an important concept in mathematics and physics.

2. How do you prove a linear operator?

To prove that a given function is a linear operator, you must show that it satisfies the properties of linearity. These properties include the preservation of addition, scalar multiplication, and composition. This can be done through direct substitution and algebraic manipulation.

3. What are the properties of a linear operator?

The properties of a linear operator include the preservation of addition, scalar multiplication, and composition. This means that when two vectors are added, the result of applying the linear operator to the sum of the vectors is equal to the sum of applying the operator to each individual vector. Similarly, when a scalar is multiplied to a vector, the result of applying the operator is equal to the scalar multiplied to the result of applying the operator to the original vector. Finally, the composition of two linear operators is also a linear operator.

4. How is a linear operator different from a regular function?

A linear operator is different from a regular function because it operates on vector spaces rather than individual numbers. This means that the input and output of a linear operator are vectors, while a regular function takes in a single number and outputs another number. Additionally, the properties of linearity must be satisfied for a function to be considered a linear operator.

5. What are some real-life applications of linear operators?

Linear operators have many real-life applications, especially in fields such as physics, engineering, and computer science. Some examples include the use of linear operators in quantum mechanics to describe the behavior of particles, in signal processing to analyze and manipulate signals, and in machine learning to perform operations on large datasets.

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