Basic Linear Algebra Proof

B^*A^*In summary, the statement being proved is that for every n × n complex matrices A and B, the conjugate of the product of A and B multiplied by a scalar α is equal to the product of the conjugate of B and the conjugate of A multiplied by the conjugate of α. The attempt at a solution involves using two "test" matrices and proving that they swap in this manner. The proof involves using the fact that the transpose of a matrix is equal to its conjugate, and then rearranging the terms to show that the conjugation property holds.
  • #1
DanielFaraday
87
0

Homework Statement



Prove the following:

For every n × n complex matrices A and B, [tex](\alpha AB)^*=\bar{\alpha }B^*A^*[/tex].

Homework Equations



None

The Attempt at a Solution



Okay, I'm just getting started on this problem. All the ideas I have come up with so far involve using two "test" matrices. The problem with this is that it doesn't prove it for any n × n matrix. Does this matter?
 
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  • #2
Ok, start small. Can you prove [tex]
(\alpha A)^*=\bar{\alpha }A^*
[/tex]

If you can, then you can just focus on proving that A and B swap like that
 
  • #3
Hmm...

This feels like trying to prove 1+1=2. It just is! I'm still working on it...
 
  • #4
Do you think this is a sufficient proof?

[tex]
(\alpha AB)^*=\bar{\alpha }\overline{AB}=\bar{\alpha }\left(\bar{A}\right)\left(\bar{B}\right)=\bar{\alpha }\left(\left(\bar{A}\right)^T\right)^T\left(\left(\bar{B}\right)^T\right)^T=\bar{\alpha }\left(A^*\right)^T\left(B^*\right)^T=\bar{\alpha }\left(B^*A^*\right)^T
[/tex]
 

1. What is Basic Linear Algebra Proof?

Basic Linear Algebra Proof is a mathematical method used to prove theorems and propositions in linear algebra. It involves using logical reasoning and mathematical properties to demonstrate the validity of a statement.

2. What are the key components of a Basic Linear Algebra Proof?

The key components of a Basic Linear Algebra Proof include defining variables, stating given information, applying relevant theorems and properties, and providing a logical argument to support the conclusion.

3. How is Basic Linear Algebra Proof different from other types of mathematical proofs?

Basic Linear Algebra Proof is specific to the field of linear algebra and utilizes its unique properties and operations. It also often involves the use of matrices and vectors, which are not typically seen in other types of mathematical proofs.

4. How can I improve my skills in Basic Linear Algebra Proof?

Practice is key to improving your skills in Basic Linear Algebra Proof. Start with simple proofs and gradually work your way up to more complex ones. It is also helpful to review theorems and properties, and to seek guidance from a knowledgeable source.

5. What are some common mistakes to avoid in Basic Linear Algebra Proof?

One common mistake in Basic Linear Algebra Proof is assuming the conclusion without providing a logical argument. It is also important to be careful with algebraic manipulations and to avoid making assumptions about the given information. Another mistake is not clearly defining variables and using inconsistent notation.

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