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

Let V be a vector space, and suppose {v_1,...,v_n} (all vectors) form a basis for V. Let V* denote the set of all linear transformations from V to R. (I know from previous work that V* is a vector space). Define f_i as an element of V* by:

f_i(a_1*v_1 + a_*v_2 + ... +a_n*v_n) = a_i

Prove that {f_1,...,f_n} gives a basis for V*.

2. Relevant equations

A basis of a subspace is a set of vectors that are linearly independent and span the subspace.

3. The attempt at a solution

By a theorem from my book, I know that for two subspaces V and W, dim V = dim W implies V = W. Since the basis for V has n vectors, dim(V) = n. Also, dim(R^n) = n. Thus, V can be equated with R^n. Since all vectors in V are in R^n, any linear transformation from V to R must have a corresponding 1xn matrix, so we know that f_1 through f_n have corresponding 1xn matrices. Since we can take the transpose of these matrices to form vectors in R^n, we can easily equate V* to R^n, so dim V* = n. This is another theorem from the book:

Let V be any k-dimensional subspace in R^n. Then any k vectors that span V must be linearly independent and any k linearly independent vectors in V must span V.

Now to finish up, I need to show that either {f_1,...,f_n} span V or they are linearly independent. I'm having trouble with either of those, though, since I don't have an explicit matrix form for any of the functions. All I know is that f_i gives the ith coordinate of a vector in V. I can prove linear independence by showing that:

c_1*[f_1] + c_2*[f_2] + ... + c_n*[f_n] = [0] implies c_1 = c_2 = ... = c_n = 0.

All I can think of from here is to multiply both sides by (a_1*v_1 + a_*v_2 + ... +a_n*v_n) to get:

c_1*a_1 + c_2*a_2 + ... + c_n*a_n = 0.

But I think I must have made an error somewhere, because we can easily show that that equation does not imply that c_1 = ... = c_n = 0.Can somebody please point me in the right direction?

Also, have I made any errors in my proof so far?

Thanks.

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# Basis of a vector space proof

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