(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm given a subspace in F^5 (not sure how to note that online) and asked to find a basis and dimension for it. I know it should be really easy, but ...

2. Relevant equations

We're given subspace W1 = {a1,a2,a3,a4,a5) in F^5: a1-a3-a4=0} .

We also know from linear algebra that dim (W) </= dim (V), and that we know the dimension from the number of vectors in the basis.

3. The attempt at a solution

From the given "constraint", if I can call it that, I can put a1 in terms of a3 and a4 such that a1 = a3 + a4. It seems that from this point, there are 3 basic variables (a1, a2, and a5) and 2 free variables (a3, a4). From this point I made a sort of matrix configuration that

(a1,a2,a3,a4,a5) = t1 (1,0,1,0,1) + t2 (0,1,0,0,0) + t3 (0,0,1,0,0) + t4 (0,0,0,1,0) where t1,t2,t3,t4 are just arbitrary parameters -- I guess coefficients of the linear combination made by these vectors.

I guess I'm confused about this sort of problem compared to the theoretical part of this math -- obviously I don't quite understand what's going on here. :uhh:

I also know that the dimension of W1 must be less than or equal to 5 (it would be 4 for the basis I made above), since it is in the vector space F^5 and dim (W1) </= dim (V). The dimension part I can figure out, it's just finding the basis that I'm lost. I guess my answer could be right, but it doesn't match what any of my classmates have, so I'm assuming this is wrong. Any help/explanation of finding bases is much needed/appreciated!

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# How to find bases for subspaces

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