Determine if vector b is a linear combination of vectors a1, a2, a3.


by roids
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roids
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#1
Feb18-13, 05:05 PM
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Hi guys. I've solved an exercise but the solution sheet says what doesn't make sense to me. Could you please help with this problem?

Determine if vector b is a linear combination of vectors a1, a2, a3.

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

b is a linear combination when there exist scalars x1, x2, x3 such that x1*a1 + x2*a2 + x3*a3 = b. right?

I put a's in a coefficient matrix and b in the augmented column. [a1 a2 a3 | b]. Row-reduced it produces a consistent system (although I get x3 a free variable - third row all zeroes). But the solution sheet says b is not a linear combination of the a vectors. Where is the catch? Should the RREF have a unique solution?

Thank you.
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tiny-tim
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Feb18-13, 05:14 PM
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hi roids! welcome to pf!
Quote Quote by roids View Post
I put a's in a coefficient matrix and b in the augmented column. [a1 a2 a3 | b]. Row-reduced it produces a consistent system (although I get x3 a free variable - third row all zeroes).
show us what you got
jbunniii
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Feb18-13, 05:24 PM
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Your solution sheet is wrong: ##2 a_1 + 3 a_2 + 0 a_3 = b##.

Simon Bridge
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#4
Feb18-13, 05:36 PM
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Determine if vector b is a linear combination of vectors a1, a2, a3.


Argh: jbunnii beat me to it.
roids
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#5
Feb18-13, 06:23 PM
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Thanks for the responses fine gentlemen. The book with the solutions is David Lay - Linear Algebra, fourth edition.

Can you please take a look at the same problem someone asked here, where the answerer said that no, b is not a linear combination? http://www.physicsforums.com/showthread.php?t=531233

While you and this someone's from Berkeley document say that b is indeed is a linear combination: http://math.berkeley.edu/~honigska/M54HW01Sols.pdf (page 6, exercise 14)

I attached the original problem and solution from the book. Can it be that the book is asking to not combine the a vectors and just test them one by one with b to see if they separately are linear combinations of b?
Attached Thumbnails
ex.PNG   ans.PNG  
jbunniii
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#6
Feb18-13, 06:54 PM
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Quote Quote by roids View Post
Can you please take a look at the same problem someone asked here, where the answerer said that no, b is not a linear combination? http://www.physicsforums.com/showthread.php?t=531233
[edit] OK, I had to re-read the thread, and actually it is agreeing with us. I'll elaborate below.
I attached the original problem and solution from the book. Can it be that the book is asking to not combine the a vectors and just test them one by one with b to see if they separately are linear combinations of b?
No, the question is clearly worded. I'm sure your interpretation was correct. The solution is simply wrong.
jbunniii
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Feb18-13, 06:58 PM
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To elaborate on what Alchemista said in this thread:

http://www.physicsforums.com/showthread.php?t=531233

He is saying that for any ##t \in \mathbb{R}##, if we set ##x_1 = 2 - 5t##, ##x_2 = 3 - 4t##, and ##x_3 = t##, then we will have ##x_1 a_1 + x_2 a_2 + x_3 a_3 = b##. Thus, not only is there a solution, there are infinitely many solutions. The solution I mentioned above in post #3 is a special case of this, with ##t = 0##.

The reason for this is that, for any ##t##,
$$-5t a_1 - 4t a_2 + t a_3 = 0$$
roids
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#8
Feb18-13, 07:21 PM
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Thanks so much jbunnii, you made my day. Love the notation by alchemista.

Have a good day!


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