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  3. let u = {(x1,x2,x3,x4) ?f4 | 2x1 = x3, x1 + x4 = 0

Let U = {(x1,x2,x3,x4) ?F4 | 2x1 = x3, x1 + x4 = 0}. (a) Prove...

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Basic Math

Let U = {(x1,x2,x3,x4) ?F4 | 2x1 = x3, x1 + x4 = 0}.

(a) Prove that U is a subspace of F4.

(b) Find a basis for U and prove that dimU = 2.

(c) Complete the basis for U in (b) to a basis of F4.

(d) Find an explicit isomorphism T : U ?F2.

(e) Let T as in part (d). Find a linear map S: F4 ?F2 such thatS(u) = T(u) for all u ? U.

Answer & Explanation Solved by verified expert
4.2 Ratings (775 Votes)
We have U x1x2x3x4 F4 2x1 x3 x1 x4 0 Then x1x2x3x4 x1x22x1 x1 aLet Xx1x22x1x1and Yy1y22y1y1 be 2 arbitrary vectors in U and let k be an arbitrary scalar F    See Answer
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