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July 7th, 2014, 07:29 AM   #1
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Proof about linear systems of equations

If $\displaystyle X_1$,$\displaystyle X_2$ are solutions of $\displaystyle AX=B \neq 0 $ then $\displaystyle aX_1 + bX_2$ is never a solution.

I tryed this way:

From the hypotesis we have $\displaystyle AX_1=B$ and $\displaystyle AX_2=B$ with $\displaystyle B \neq 0$. Then:

$\displaystyle A(aX_1 + bX_2)=aAX_1 + bAX_2=aB + bB=B$ which is true only if $\displaystyle a+b = 1$.

From what I found a linear combination can be a solution so I would like to know what is wrong with what I did.
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July 7th, 2014, 01:01 PM   #2
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Looks O.K.
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July 25th, 2014, 06:12 PM   #3
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Hi, trying to learn Linear Algebra and found this very interesting because the proof seems to contradict everything I'm learning, but the proof clearly shows why the sum of the solutions most likely will not be a solution.

My reasoning is this: since there's more than one solution, there's an infinity of solutions. These solutions must form a subspace, and by the definition of subspace, addition of any two vectors must lie within the subspace. Therefore, the sum is also a solution.

Now as I'm writing this, I'm realizing that also by definition of subspace, the zero vector is in it, and the zero vector would make B = 0, which contradicts the condition in the question. So the solutions must not be coming from a subspace. Is that really the reason then why the sum of two solution vectors is not a solution? Just trying to understand this conceptually. I would think that adding two vectors from an infinite line will land you on the same infinite line... Is that only true for a line passing through 0?
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