# 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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Expert Answers

tiburtius | Certified Educator

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)`.