Lin Alg - Matrix multiplication (Proof by contrapositive)

In summary, the conversation discusses proving that A = B if AX = BX for all n-tuples X, using the contrapositive method. The proof provided is considered sufficient, but it is noted that proving directly might be more transparent.
  • #1
mattmns
1,128
6
Hello, here is the question my book is asking:

Let A, B be two m x n matricies. Assume that AX = BX for all n-tuples X. Show that A = B.
-------

So I decided to try and prove the contrapositive, which is (unless I am mistaken): If [itex]A \neq B[/itex], then there is some X such that [itex]AX \neq BX[/itex]

Proof:

Assume [itex]A \neq B[/itex]
Then [itex]A^j \neq B^j[/itex] for some j, where [itex]A^j, B^j[/itex] are the j-th columns of A and B.
Then, let [itex] X = E^j[/itex] be the unit vector with 1 in the j-th spot, the same j where [itex]A^j \neq B^j[/itex]
So [itex]AX = AE^j = A^j[/itex] and
[itex] BX = BE^j = B^j[/itex]
and so [itex]AX \neq BX[/itex] for [itex]X = E^j[/itex] as [itex]A^j \neq B^j[/itex]

Thus if [itex]A \neq B[/itex] there is some X such that [itex]AX \neq BX[/itex]
So, as the contrapositive is logically equivalent, we have just showed that if AX = BX for all X, then A = B. Where A, B are two m x n matricies, and X is an n-tuple.
------

First, is the contrapositive correct, and if so then is the proof correct. The whole thing looks perfectly sufficient to me. Thanks!

Also, I just realized that it is very easy to just prove it directly, but I am still curious to see if this is a sufficient proof. Thanks!
 
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  • #2
[tex]test\pi[/tex] [tex]$test\pi$[/tex]
 
  • #3
I would call that sufficient.
 
  • #4
The proof is good. Proving directly might be more transparant though. You already noted the crucial [itex]AE^j=A^j[/itex], so if [itex]AX=BX[/itex] for all X, then [itex]A^j=AE^j=BE^j=B^j[/itex] for j=1,2,...,n. So A=B.
 
  • #5
Yep, I saw that just as I was posting my proof. Thanks.
 

1. What is matrix multiplication?

Matrix multiplication is an operation in linear algebra that combines two matrices to produce a new matrix. It is defined as the product of each element in the first matrix with the corresponding element in the second matrix, and then summing the results.

2. What is a proof by contrapositive?

A proof by contrapositive is a method of proving a statement by showing that its negation leads to a contradiction. It involves assuming the opposite of the statement and then showing that it leads to a logical inconsistency, thereby proving the original statement to be true.

3. How is matrix multiplication related to the proof by contrapositive?

In the context of linear algebra, matrix multiplication can be used to prove statements by contrapositive. By multiplying two matrices and obtaining a contradiction, we can prove that the original statement is true. This approach is commonly used in proofs involving matrix equations and transformations.

4. Can you provide an example of a proof by contrapositive using matrix multiplication?

Sure, let's say we want to prove that if A and B are invertible matrices, then AB is also invertible. We can do this by assuming that AB is not invertible and then showing that this leads to a contradiction. By the definition of invertibility, we know that AB is not invertible if and only if its determinant is equal to 0. So, we can multiply the determinants of A and B, and if the result is 0, then AB is not invertible. However, this would contradict the fact that both A and B are invertible, since their determinants are non-zero. Therefore, our assumption was false and AB must be invertible.

5. Are there any other applications of matrix multiplication in mathematical proofs?

Yes, matrix multiplication is a fundamental operation in linear algebra and has numerous applications in mathematical proofs. It is commonly used in proofs involving systems of linear equations, eigenvalues and eigenvectors, and linear transformations. Additionally, matrix multiplication is closely related to other concepts in mathematics such as determinants, matrices as linear operators, and vector spaces.

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