This is David C lays book for linear algebra 3rd edition section 2.1 problem 15

Question

This is David C lays book for linear algebra 3rd edition section 2.1 problem 15 part B on page 117.

The Question is:

Each Column of AB is linear combination of the columns of B using weights from the corresponding columns of A. I agree this is wrong however the given answer reverse A and B in the above statement as the correct answer. However it seems to me that the correct answer should be as follows

Each column of AB is a linear combination of the rows of A using weights from the corresponding columns of B.

The difference here is the rows of A where the book and the answer on the site both state the columns of A.

I come to this conclusion since each row of A is effectively multiplied by the columns of B.

for example

A= a  b  c

     d  e  f

B= g  h

     i   j

    k  m

Thus the first column of AB = ag+bi+ck dh+ej+jm

Thus the coloums of AB are made up of the componets of the rows of A and the columns of B.
So can someone explain to me what is the correct answer and why.

Thanks
Daniel

Explanation / Answer

This is David C lays book for linear algebra 3rd edition section 2.1 problem 15 part B on page 117.

The Question is:

Each Column of AB is linear combination of the columns of B using weights from the corresponding columns of A.

THIS IS A STATEMENT ..NOT A QUESTION!!!

SO WHAT IS THE QUESTION

TO SAY WHETHER IT IS TRUE OR FALSE ??..ASSUMING SO ..

I agree this is wrong however the given answer reverse A and B in the above statement as the correct answer.

OK..THIS IS WRONG ...

However it seems to me that the correct answer should be as follows

Each column of AB is a linear combination of the rows of A using weights from the corresponding columns of B.

WHAT IS CORRESPONDING?

THE FACT IS

COL.1 OF AB IS A LINEAR COMBINATION OF ROW 1 OF A WITH COL.1 OF B AS ITS WEIGHTS.

OR ...WE CAN ALSO SAY IT AS.....

COL.1 OF AB IS A LINEAR COMBINATION OF COL. 1 OF B WITH ROW 1 OF A AS ITS WEIGHTS.

AND SIMILARLY ...

COL.2 OF AB IS A LINEAR COMBINATION OF ROW 2 OF A WITH COL.2 OF B AS ITS WEIGHTS.

OR ...WE CAN ALSO SAY IT AS.....

COL.2 OF AB IS A LINEAR COMBINATION OF COL. 2 OF B WITH ROW 2 OF A AS ITS WEIGHTS.

The difference here is the rows of A where the book and the answer on the site both state the columns of A.

SO AS YOU CAN SEE ABOVE THAT WE CAN EXPRESS THE SAME FACT IN 2 WAYS , BUT THE INVARIANT THING

IS THAT IF ROW IS TAKEN AS WEIGHTS , THEN THEY ACT ON COL.ELEMENTS

AND IF WE TAKE COLUMN AS WEIGHTS , THEN THEY ACT ON ROW ELEMENTS.

I come to this conclusion since each row of A is effectively multiplied by the columns of B....OK

for example

A= a b c

   d e f

B= g h

     i   j

    k m

Thus the first column of AB = ag+bi+ck dh+ej+jm

Thus the coloums of AB are made up of the componets of the rows of A and the columns of B.
So can someone explain to me what is the correct answer and why.

HOPE THE ABOVEEXPLN. IS OK WITH YOU ..

Thanks
Daniel

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