This is David C lays book for linear algebra 3rd edition section 2.1 problem 15
ID: 1942419 • Letter: T
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