I'm given this set of vectors:
$$(-a,1,b),(a,1,2a),(0,1,2a),(2a,-1,a+2)$$
EDIT: I actually need to figure out for what values of $a$ and $b$ the first vector is a linear combnation of the other three. After some comments I noticed that the question was better phrased this way.
I tried to see when it was not a linear combination, by putting them in the columns of a matrix and performing elementary transformations on it to try and put it in echelon form ($AX=0$, where A is that matrix and X is the vector of the coefficients of the linear combination) but I'm not being able to solve it no matter what I try; plus, I feel like there be a smarter way to solve this. Can someone help me out?
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Lourenco Entrudo
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You may check this "https://www.sciencedirect.com/topics/mathematics/linearly-dependent" – Jan 15 '22 at 13:31
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Hint : Can these 4 vectors be linearly independant together ? – Lelouch Jan 15 '22 at 13:32
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@Lelouch they can – Lourenco Entrudo Jan 15 '22 at 13:35
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1So you can find a set of 4 linearly independent vectors of $\mathbb{R^3}$ ? – Lelouch Jan 15 '22 at 13:36
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You're right. But I can values of a and b for which the first one is independent from the other three – Lourenco Entrudo Jan 15 '22 at 13:40
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Can you have $;n+1;$ or more l.i. vectors in an $,n,-$ dimensional space? – DonAntonio Jan 15 '22 at 13:40