The question #5 The set of all functions such that f(0) = 0 The set of all funct

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The question #5

The set of all functions such that f(0) = 0 The set of all functions such that f(-1) = f(1) = 1 The set of all functions such that f is infinity differentiable at all Prove that the solution space of y" = 0 is spanned by t and 1. Prove that v = {0,-3,4} and w = {1,-1,0} span the same subsapce of R3 and s = {-1, 4, -4} and u = {1, 2, -4}. (Hint: Show that any linear combination of the vectors v and w can be written as a linear combination of s and u and vice versa.) Prove that the functions sin2x, cos2x, and sinxcosx span the same subspace as 1, sin2x, cos2x.

Explanation / Answer

conside subspace spanned by v and w it is a{0,-3,4}+b{1,-1,0} = {b,-3a-b,4a} let b = -c+d and a = -c -d so then -3a-b is 4c+2d so {b,-3a-b,4a} = {-c + d ,4c+2d , -4c-4d} =c{-1,4,-4} + d{1,2,-4} so v,w and s,u span the same subspace

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