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Determine the basis for each of the following subspaces of `R^3` . Give the dimension...

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tobydator | Student, College Freshman | Honors

Posted August 24, 2013 at 10:57 AM via web

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Determine the basis for each of the following subspaces of `R^3` . Give the dimension of each subspace.

a.(a,a,b)

b.(a,b,a+b)

Please show the solution... Thank you:))

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted August 29, 2013 at 12:06 PM (Answer #1)

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Let's assume that `a` and `b` are linearly independent vectors from `R^3` (they do not lie on the same line), that is there is no such real number `lambda` such that `a=lambda b.`

a. The basis of this space is `(a,b)` because `a` and `b` are independent by assumption and two `a`'s are the same vector so they are dependent. Also because there are two vectors in the basis dimension is 2.

b. Again the basis is `(a,b)` because `a` and `b` are independent by assumption and vector `a+b=1cdot a+1cdot b` is linear combination of those two vectors so it's in the same space. Again as in a. dimension is 2.

We could have chosen a different basis e.g. `(a,a+b)` or  `(b,a+b)`.

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