Simple Proofs for Matrix Algebra Properties: A Beginner's Guide

[leehufford]

Hello,

So I am struggling with a couple very simple proofs of properties of matrix algebra. This is the first time I have ever had real proofs in math (Linear algebra). For the first one, I have it from our text but need a little help, and I am completely lost on the second one.

1) Prove that for matrices A and B that

A + B = B + A

We must show that the entries are identical for each. Therefore

(A+B)_ij = (B+A)_ij

= A_ij + B_ij (Definition of addition??)
= B_ij + A_ij (Commutativity of real numbers)
= (B+A)_ij (Definition of addition??)

So I totally get step 2. We're proving the commutativity of matrices, so we are allowed to use the commutativity of real numbers. But when the book says "definition of addition" it seems like they mean to say "distributivity"... so this is throwing me off.

Proof number 2 is:

Prove that (c+d)A = cA + dA where c,d are scalars and A is a matrix.

The only thing I can think to do is assume that their entries are equal, like in the first one, but then I am not sure where to go from there. So in summary,

1) Why does the book say "definition of addition" when it does and,
2) What is the first/second step for proof #2?

The book really doesn't provide any strategies for proofs, it seems like every proof is different at this point. I'm just not "seeing" it yet. Thanks so much in advance,

-Lee

 

[mathman]

Definition of addition: think of 1+2 = (1+2) = 3. It seems hard to explain further.

 

[leehufford]

Definition of addition: think of 1+2 = (1+2) = 3. It seems hard to explain further.

I was having a hard time with the equivalence of (A+B)_ij = A_ij + B_ij.

While previewing my post I think I got it. Because addition of matrices is defined component wise the statement is true. Its the component wise definition of matrix addition here. Its a little more than (1 + 2) = 3 right? Did I explain it further?

Any advice for proof #2?

Thanks for the reply.

Lee

 

[AlephZero]

I think it would have been better if the book said "definition of matrix addition", not just "definition of addition".

You have the right idea for #2. It can take a bit of practice to "see" how to write out this type of proof formally, though.

Remember A is a matrix, but A_ij is just a number. You know how to do arithmetic with numbers. You are trying to prove things about doing arithmetic with matrices. So you need to break the matrix operation down into numbers, do the arithmetic, and then convert the result back into a matrices.

I would start with
[ (c+d)A ]_ij
= (c+d)A_ij (definition of scalar multiplication)
etc

 

[blue_raver22]

Hello Lee,

First of all, it's great that you are working on proofs in linear algebra. It can be challenging at first, but with practice and understanding of the concepts, it will become easier.

To answer your first question, the "definition of addition" in the first proof refers to the definition of addition for matrices, which is that the sum of two matrices is obtained by adding the corresponding entries. This is different from the distributivity property of real numbers, which is used in step 2. So, in step 3, we are using the definition of addition for matrices to show that the entries are identical on both sides.

Moving on to the second proof, here is a possible approach:

1) Start with the left-hand side: (c+d)A = [(c+d)Aij]
2) Use the definition of scalar multiplication for matrices: [(c+d)Aij] = cAij + dAij
3) Rearrange the terms to match the right-hand side: cAij + dAij = (cA + dA)ij
4) Use the definition of addition for matrices to show that the entries are identical on both sides.

I hope this helps. Remember, the key to solving proofs is to understand the definitions and properties, and then apply them logically. Keep practicing and you will get better at it. Good luck!

Best,