Linear Algebra: Equivalence of Linear Transformations

In summary, two linear transformations B and C are equivalent if and only if there exist invertible linear transformations P and Q such that PB = CQ. Additionally, if A and B are equivalent, then so are their dual transformations A' and B'. It is possible for linear transformations A and B to be equivalent, but for A^2 and B^2 to not be equivalent. It is also possible for a linear transformation A to be equivalent to a scalar a, but not equal to a. The definition of equivalence between A and B is that there exist linear transformations E and F such that B = E^-1AF.
  • #1
Fringhe
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Homework Statement


1) two linear transformations B and C are equivalent iff there exist invertible linear transformations P and Q such that PB=CQ
2) if A and B are equivalent then so are A' and B' in dual space
3) Do there exist linear transformations A and B such that A and B are equivalent but A^2 and B^2 are not?
4) Does there exist a linear transformation A such that A is equivalent to a scalar a but A is not equal to a?

The Attempt at a Solution


I really don't know where to start. I know that if two l.ts. A and B are equivalent then (AB)^-1 = B^-1A^-1. But that's where I am now.
 
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  • #2
Ok for the first question, two lts B and C are equivalent iff there exist lts E and F such that
B = E^-1 C F
Now let E = P and let F=Q, we have
B= P^-1 C Q or PB = CQ so this means that the lts P and Q must be invertible?
 
  • #3
Can you please repeat your definition for equivalence between A and B? I'm not sure I follow.
 

1. What is the definition of equivalence in linear algebra?

Equivalence in linear algebra refers to two matrices being equal in terms of their row operations. This means that the two matrices have the same number of rows and columns, and each element in one matrix can be transformed into the corresponding element in the other matrix through a series of elementary row operations.

2. How do you determine if two matrices are equivalent?

To determine if two matrices are equivalent, you can perform elementary row operations on one matrix to transform it into the other. If this is possible, then the two matrices are equivalent. Additionally, you can also check if the two matrices have the same number of rows and columns and if each element in one matrix is equal to the corresponding element in the other matrix.

3. Can equivalent matrices have different solutions to a system of linear equations?

No, equivalent matrices will always have the same solutions to a system of linear equations. This is because elementary row operations do not change the solutions to a system of linear equations, as they only change the representation of the equations.

4. How is the concept of equivalence used in solving linear equations?

Equivalence is used in solving linear equations by allowing us to transform a system of equations into an equivalent system that is easier to solve. This is done through elementary row operations, which help to eliminate variables and simplify the system of equations.

5. Are all equivalent matrices equal?

Yes, all equivalent matrices are equal. This is because the definition of equivalence in linear algebra states that two matrices are equivalent if they can be transformed into each other through elementary row operations, which do not change the values of the elements in the matrices. Therefore, equivalent matrices will always be equal to each other.

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