Proving the Formula for Matrix Multiplication | Homework Statement & Equations

[Shaun Culver]

Homework Statement

Prove the formula.

Homework Equations

Matrix multiplication:
$$(\text{AB})_{i \,j}=\sum _{k=1}^n a_{i \,k}b_{k \,j}$$

The Attempt at a Solution

I do not know how to "prove" the formula for arbitrary values of $k$ and $n$.

 

[statdad]

Well,

$$(\text{AB})_{i \,j}=\sum _{k=1}^n a_{i \,k}b_{k \,j}$$

is the very definition of matrix multiplication, is it not (see
http://en.wikipedia.org/wiki/Matrix_multiplication)

What do you mean by prove?

 

[Shaun Culver]

On page 36 of "A Course in Modern Mathematical Physics" by Peter Szekeres, the matrix multiplication rule is given, and then an exercise follows immediately after the rule stating: "Prove this formula".

Please see: http://books.google.co.uk/books?id=...a=X&oi=book_result&resnum=1&ct=result#PPR5,M1

 

[HallsofIvy]

Okay, what he is doing is defining the matrix corresponding to a linear transformation, then defining the multiplication of two matrices as the matrix corresponding to the composition of the two corresponding linear transformation, finally giving that formula. What is asked here is that you show that this formula really does give the matrix corresponding to the composition of two linear transformations. I would recommend that you look at what the linear transformations and the two matrices do to each of the basis vectors in turn.