# Thread: Linear equations - Summation convention

1. ## Linear 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.

2. ## Re: 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

$[AB]_{il}=\sum_k a_{ik}b_{kl}$ (1)

and

$[BC]_{kj}=\sum_l b_{kl}c_{lj}$ (2)

So we get

\begin{align*}[(AB)C]_{ij} &= \sum_l\left(\sum_k a_{ik}b_{kl}\right)c_{lj} && \text{by (1)}\\ &=\sum_l\sum_k a_{ik}b_{kl}c_{lj}&& \text{by distributivity}\\ &=\sum_k\sum_l a_{ik}b_{kl}c_{lj}&& \text{by exchanging the order of summation, i.e., commutativity}\\ &=\sum_k a_{ik}\left(\sum_l b_{kl}c_{lj}\right)&& \text{by distributivity}\\ &=[A(BC)]_{ij}&& \text{by (2)} \end{align*}

3. ## Re: Linear equations - Summation convention

Hi, emakarov.