Problems in Tensors
 

Here is an example that is solved in Vector & Tensor Analysis by Dr Nawazish Ali Shah. Example 2 of Chapter 7 is aixj+akyk. Evaluate it.
In my opinion here are two dummy indices so it is a double summation.
I think if v assume i as rows and k as columns then v vl evalute it as
a1x1+a1y1 + a1x1+a2y2 + a1x1+a3y3
+ a2x2+a1y1 + a2x2+a2y2+ a2x2+a3y3
+a3x3+a1y1 + a3x3+a2y2+ a3x3+a3y3
As i and k are two different indices. The book, however, has written the result as a1x1+a1y1 + a2x2+a2y2 + a3x3+a3y3
I think for the above given result the question in book should be aixi+akxk.
Any one having any idea about the tensor notations kindly sort out the problem.

You ve complicated it a bit.

It has to be summed over j and k. Now in cartesian coordinates, these indices take values of 1, 2 and 3. Replace the index of j and k accordingly and you get a1x1+a1y1 + a2x2+a2y2 + a3x3+a3y3.

You ve confused things by introducing rows and columns which is wrong. Just replace the indices with these values and if there is dummy index, sum over that. Here it is a simple expansion.