(Solution) - Let V be a vector space A subset of the -(2025 Original AI-Free Solution)

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Let V be a vector space. A subset of the form A = {w + a | w ? W} where V ? V is a subspace and a ? V is a fixed vector is known as an affine subspace of V.
(a) Show that the affine subspace A ? P is a genuine subspace if and only if a ? W.
(b) Draw the affine subspaces A ? R2 when
(i) W is the x-axis and a = (2, l)T
(ii) VP is the line y = 3/2x and a =(1,1)T.
(iii) w is the line {(r, - t)T |t ? R}, and a = (2, -2)T.
(c) Prove that every affine subspace A ? R2 is either a point, a line, or all of R2.
(d) Show that the plane x - 2y + 3z = 1 is an affine subspace of R3.
(e) Show that the set of all polynomials such that p(0) = 1 is an affine subspace of P(n)