Hi, i want to show that zi=cij*xj, where cij=aik*bkj, if yi=bij*xj and zi=aij*yj. (i,j,k are subscripts).

I have been struggling with this for a week. Any help is appreciated, thanks in advance.

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- October 9th 2013, 01:21 PMFilipVzLinear equations - Summation convention
Hi, i want to show that zi=cij*xj, where cij=aik*bkj, if yi=bij*xj and zi=aij*yj. (i,j,k are subscripts).

I have been struggling with this for a week. Any help is appreciated, thanks in advance. - October 9th 2013, 02:18 PMemakarovRe: Linear equations - Summation convention
In matrix form, you want to show that y = Bx and z = Ay implies that z = (AB)x. From the assumption, it follows that z = A(Bx), so what you need to show is associativity of matrix multiplication: A(Bx) = (AB)x.

Let A, B and C be matrices of compatible sizes. Then

(1)

and

(2)

So we get

- October 9th 2013, 02:29 PMFilipVzRe: Linear equations - Summation convention
Hi, emakarov.

Thank you for the answer.

Is there any other way (more trivial) to prove that? - October 9th 2013, 02:51 PMemakarovRe: Linear equations - Summation convention
I am not sure. One can try writing the sums explicitly instead of using ∑. You can also try becoming more comfortable with ∑-notation and see what distributivity and exchanging limits means when written explicitly. Finally, you may search for explanations of the associativity of matrix multiplication; there must be good descriptions. (I am not sure if you realized that this problems boils down to associativity.)

Feel free to post here what you tried or found.