1) let V be the vector space of all continuous functions on the interval [0,1] a
ID: 1943902 • Letter: 1
Question
1) let V be the vector space of all continuous functions on the interval [0,1] and letWa := {f ? V : f(1) + f(0) = a} ? V. Find the values of the parameter 'a' for which Wa is a linear subspace of V. Prove your answer, and demonstrate that Wa is not a linear supspace of V for all other values of 'a'
2)Let V be a vector space, and let U, W ? V be its linear subspaces. Define
Z = {2u + 3w : u?U, w?W}
Prove that Z is a linear subspace of V.
3) Let p,q,r be linearly independent vectors in some vector space V. Define u := p-2q.
a)are u,q,r linearly independent?
b)are u,p,q linearly independent?
in both cases, prove your answer.
Explanation / Answer
a)Let f,g V
f(1)+f(0) = a
g(1)+g(0) = a
[f(1)+f(0)]+[g(1)+g(0)] = a + a = 2a
For Wa be subspace a = 2a => a = 0
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