Proving Matrix Addition/Scalar Multiplication Properties

[asdf1]

just curious,
how do you prove the basic matrix addition and scalar multiplication properties if you don't want to prove it just by giving examples?

 

[quasar987]

Giving examples doesn't prove anything. Suppose you gave 10 billion exemples of where a theorem was true. Then there's nothing in that which guarentees that the billion and one-th exemple won't fail!

As for your question, matrix multiplication and addition are not provable! They are defined. First we define matrix as a table of numbers, then we define addition on them and mult. by a scalar. Then we go on finding what are the properties given such definitions. These are the theorems of matrix theory.

 

[fourier jr]

just curious,
how do you prove the basic matrix addition and scalar multiplication properties if you don't want to prove it just by giving examples?

what exactly are you trying to prove? you can prove, say, commutativity of addition by using the definition. that is, you do it by taking (any! that's the key) mxn matrices A & B, and adding the individual entries together like the definition says. so at the end you get A+B=B+A. is that the sort of stuff you want to prove?

 

[asdf1]

yes, just like your example A+B=B+A, how are you supposed to prove that? because the only way i can think of is by giving examples, because A and B can be of any size, which makes it hard to be generalized ( i think~)...

 

[TD]

If you know how matrix addition is defined (adding the equivalent elements), then it follows naturally from the commutativity of the ordinary addition, which says that a+b = b+a (scalairs).

 

[asdf1]

hmm... I've heard that you can prove it with special sigma thingys that physicts use? or that physicts can use some special type of notation or something for those kinds of proofs?
@@a

 

[TD]

If you mean permutation (sigma), it is used to define determinants for example.

 

[HallsofIvy]

The way you prove things is by using the definitions! The precise statement of the definitions!

For example, If A= [a_ij] is a matrix and B= [b_ij] is another matrix

The definition of A+ B is [(a_ij+ _ij]- each entry in A+B is the sum of the corresponding entries of A and B separately.

Now, using that same definition, what is B+A?

Remember that the entries of A and B are real numbers and addition of real numbers is commutative.

 

[quasar987]

Oops, I misunderstood your question asdf1, sowwy 'bout that. Had I read it right, I would have said what HallsofIvy said, and added to your confusion...

[...]because the only way i can think of is by giving examples, because A and B can be of any size, which makes it hard to be generalized ( i think~)...

...that matrix addition is only defined for matrices A and B of the same size, so don't even try to add matrices of different size, the laws of addition of those matrices is not defined.

 

[teclo]

i've recently had to prove many of the properties of matrix operations. associative, etc...
the way i attempted to do it was either algebraicaly or geometrically.

for the most part it's involved pages of obnoxious vectors a,b,c broken into cartesian coordinates. I'm not so good at the geometric business, drawing three dimensional stuff doesn't seem to be my forte.

 

[asdf1]

can you prove that kind of stuff by using permutation?