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# Subspace of R^4 watch

1. Let u1,u2,u3 be three linearly independent vectors in R^4.

If u4 is another vector in R^4 that does not lie in Span {u1,u2,u3} show that u1,u2,u3,u4 are linearly independent.

Identify the SubSpace Span{u1,u2,u3,u4} of R^4

Ok, so I know that you require 4 vectors to form a basis of R^4 and given that u4 does not lie in span(u1,u2,u3), it is not a linear combination of those vectors and hence must be the fourth vector in R^4 which is also linearly independent. I'm not sure how to prove this though?

I have no idea about the second question. What exactly is it asking for?
2. (Original post by Zilch)
Ok, so I know that you require 4 vectors to form a basis of R^4 and given that u4 does not lie in span(u1,u2,u3), it is not a linear combination of those vectors and hence must be the fourth vector in R^4 which is also linearly independent.
A fourth vector - not the fourth vector.

I'm not sure how to prove this though?
Just write out the full definition of linear independence and then write out exactly what elements in the span look like and the proof is then instant. As with your other question - what is holding you back is the ability to write out as a precise mathematical statement, the definition of linear independence.

I have no idea about the second question. What exactly is it asking for?
Write out what the span of those four vectors is i.e. the set of vectors blah blah such that x,y,z

Once you do, that, then you can say that this is precisely equal to some familiar vector space.
3. (Original post by Mark85)
A fourth vector - not the fourth vector.

Just write out the full definition of linear independence and then write out exactly what elements in the span look like and the proof is then instant. As with your other question - what is holding you back is the ability to write out as a precise mathematical statement, the definition of linear independence.

Write out what the span of those four vectors is i.e. the set of vectors blah blah such that x,y,z

Once you do, that, then you can say that this is precisely equal to some familiar vector space.
Got it thanks.!

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