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Mariah's avatar

Did my textbook make a mistake, or am I just misunderstanding this concept (linear algebra)?

Asked by Mariah (24428points) September 17th, 2012

This is not homework; I am studying for an exam.

My textbook has a practice problem: “determine if b is a linear combination of a1, a2, and a3.”

Please note that all of the following matrices are supposed to be vertical, with 1 column and 3 rows, but of course the limitations of Fluther are such that I can’t write them that way.

a1 = [1 -2 0], a2 = [0 1 2], a3 = [5 -6 8], b = [2 -1 6]

I wrote the augmented matrix with a1, a2, a3, b as the columns and put it in reduced row echelon form. Here is my result (this cannot be where my mistake is because I had my calculator do it for me):

1 0 5 2
0 1 4 3
0 0 0 0

It is my understanding that because a solution exists (in fact, many solutions because x3 is free), b is a linear combination of a1, a2, a3. But my textbooks disagrees! Am I thinking about this wrong, or is my textbook wrong?

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5 Answers

Response moderated (Unhelpful)
LostInParadise's avatar

The book is wrong and it is easy to demonstrate it. Let the coefficients of a1, a2 and a3 be X1, X2 and X3. From the first equation we have x1 = 2 – 5*X3 and from the second equation we have X2 = 3 – 4X3. We can make X3 anything we want. Let X3 = 1. Then X1 = 3 and X2 = -1. You can see directly that b = 3*a1 -a2 + a3

Mariah's avatar

Thank you, @LostInParadise, this is what I thought to be the case, but was not confident in my opinion over that of the textbook. Very much appreciated.

PhiNotPi's avatar

@LostInParadise is correct. Textbooks do have errors. Even if the textbook makes an error less than 1% of the time, chances are it contains more than 100 answers.

Response moderated (Spam)

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