Solving a Problem in My Assignment: X1, X2, and X3

AI Thread Summary
The discussion revolves around solving a problem in an assignment involving vectors X1, X2, and X3. The user is uncertain about the correct values for X3, considering two possibilities based on the difference between X1 and X2. It is established that the kernel of matrix A is at least one-dimensional, indicating multiple solutions exist. The only confirmed solutions are of the form X1 + λ(X2 - X1), but it's unclear if these encompass all possible solutions. The conversation highlights the complexity of the problem and the need for further clarification on the parameters involved.
Soma
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Homework Statement
Let X1 = [(1,2,3)] and Let X2 = [(4,5,6)] be two solutions of the linear system AX = B. Find all solutions X3 of this system, such that X3 ≠ X1 and X3 ≠ X2.
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This is just a small part of a question I have in my assignment and I'm not sure how to solve it, nothing in my eBook or our presentation slides hints at a similar problem, what I tried was I noticed that X1 and X2 have the difference of (3,3,3) and I assume either X3 = (3,3,3) or X3 = (7,8,9) is that right or am I getting it wrong?
 
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There should be infinitely many possibilities. We know nothing about ##A## or ##B##. The only fact we have is what you have already observed: ##Ax_1=b=Ax_2 \Longrightarrow A(x_2-x_1)=A(3,3,3)=0##. This means the kernel of ##A## is not trivial and at least one dimensional. Thus the kernel may have all dimensions ##1,2,3## depending on what ##A## and ##b## are.

We only know for sure that all vectors ##x_1+\lambda (x_2-x_1)## are solutions, but we cannot know whether these are all solutions.
 
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Thank you so much for the help! I wasn't sure what to do at first but that makes a lot of sense
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
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