- #1
danago
Gold Member
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Given a set of four vectors in space, prove that at least one is a linear combination of the other three.
I think i am able to visualize this one in my head, but don't really know how to write a solid proof for it.
Lets say i take any three of the vectors; The three will be either coplanar or non-coplanar. If they are coplanar, then any of those three vectors can be represented as a linear combination of the other two. If they are non-coplanar, then any vector in R^3 i.e. the fourth vector, can be be written as a linear combination of the three.
Now i think i have visualised it correctly, but it certainly doesn't feel like i have proven much. Can anyone suggest how i could go about doing so?
Thanks,
Dan.
I think i am able to visualize this one in my head, but don't really know how to write a solid proof for it.
Lets say i take any three of the vectors; The three will be either coplanar or non-coplanar. If they are coplanar, then any of those three vectors can be represented as a linear combination of the other two. If they are non-coplanar, then any vector in R^3 i.e. the fourth vector, can be be written as a linear combination of the three.
Now i think i have visualised it correctly, but it certainly doesn't feel like i have proven much. Can anyone suggest how i could go about doing so?
Thanks,
Dan.