Linear algebra problem no.1

[mathwannabe]

Hello everybody

I am doing a course in linear algebra. I was forced to skip some of the classes in linear algebra and I am trying to get a grip on it by studying from the books. The book I am studying from only provides definitions, theorems and proofs and, at the end of each section, problems for practice. It does not provide answers to those problem nor does it provide examples for solving problems. So, here is what is troubling me:

1) If \(\displaystyle a\) and \(\displaystyle b\) are fixed vectors from a given vector space \(\displaystyle V\), examine if there are, in the given vector space \(\displaystyle V\), such vectors \(\displaystyle u\) and \(\displaystyle v\) for which \(\displaystyle 3u+2v=a\) and \(\displaystyle 4u+3v=b\)

So, as I have mentioned, I have skipped some of the classes, but I have managed to work something out on my own. What is troubling me is that I have no idea wheather I have done this correctly. Here, I am providing my work:

First, I assigned to each component of the given vectors a uniqu letter:

\(\displaystyle a=(n,m)\)
\(\displaystyle b=(p,q)\)
\(\displaystyle u=(x,y)\)
\(\displaystyle v=(z,t)\)

Then, I constructed the following:

\(\displaystyle 3u+2v=a\)
\(\displaystyle 3(x,y)+2(z,t)=(n,m)\) (1)
\(\displaystyle (3x,3y)+(2z,2t)=(n,m)\)

and

\(\displaystyle 4u+3v=b\)
\(\displaystyle 4(x,y)+3(z,t)=(p,q)\) (2)
\(\displaystyle (4x,4y)+(3z,3t)=(p,q)\)

From that, I got two systems with two variables:

\(\displaystyle 3x+2z=n\)
\(\displaystyle 4x+3z=p\)

and:

\(\displaystyle 3y+2z=m\)
\(\displaystyle 4y+3z=q\)

I solved for each variable, and got:

\(\displaystyle x=3n-2p\)
\(\displaystyle y=3m-2q\)
\(\displaystyle z=3p-4n\)
\(\displaystyle t=3q-4m\)

I switched those values back to (1) and (2) and after some cleanup finally got \(\displaystyle (n,m)=(n,m)\) and \(\displaystyle (p,q)=(p,q)\), so my answer is that there are vectors \(\displaystyle u,v\) that meet the given conditions.

Is this even correct? is this the "right" way to do it, if it is correct? Is there some generalized way for dealing with this types of problems?

 

[HallsofIvy]

You are assuming that V is a two dimensional vector space and I can see nowhere that given. And, indeed, most of what you did is completely unecessary. You are given the two equations 3u+ 2v= a and 4u+ 3v= b. Solve those two equations for u and v in terms of a and b. For example if multiply the first equation by 4, multiply the second equation by 3, and subtract. What do you get? Or, multiply the first equation by 3, mutiply the second equation by 2, and subtract. What do you get?