MATH 235 W22 - A3
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MATH 235 W22 - A3
Question 1
(A) Let V = ,╱ 、b(a) : a, b e f、 with addition e defined by ╱ 、b(a) e ╱ 、d(c) = 、,
and let c = è with multiplication o defined by α o ╱ 、d(c) = ╱α(α)d(c)、, where αc is the
product of the complex number α and the real number c.
Show that (V, e, c, o) is not a vector space.
(B) Let V = ,╱ 、b(a) : a, b e è、 with addition e defined by ╱ 、b(a) e ╱ 、d(c) = 、,
and let F = f with multiplication o defined by α o ╱ 、d(c) = ╱α(α)d(c)、, where αc is the
product of the real number α and the complex number c.
(i) Show that (V, e, c, o) is a vector space.
(ii) Show that ,╱ 、1(1) , ╱ 、、i(i) is linearly independent in this vector space.
(iii) Find a basis for this vector space, and thus state the dimension of (V, e, F, o).
Question 2
Consider the vector space V = C时 (f).
Find a basis BWi for the following subspaces Wi of V using vectors in the subspace Wi , for i = 1, 2, 3.
(a) W1 = Span({1, x, x2 l).
(b) W2 = Span( ,cos(2x), sin2 (x), cos2 (x)、).
(c) W3 = Span({1, x, cos(x), cos(2x)l).
2022-02-16