Exercise 14. Let A be an affine space, B be an affine subspace of A, and...

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Exercise 14. Let A be an affine space, B be an affine subspace of A, and p be a point in B. 1) Show that the pointed space (B, p) is linear subspace of the vector space (A, p). So (B, p) is a vector space as well. 2) Show that B is an affine space modeled on vector space (B, p). (This says that affine subspaces are affine spaces.) 3) Let T: VW be a linear map, w be a vector in W such that T-1 (w) := {ve V | T(v) = w} is a nonempty set. Show that T-1(w) is an affine subspace of the affine space V ( i.e., Vaff), but not a linear subspace of the vector space V unless w = 0. (This says that taking inverse image of a linear map will get us out of the realm of vector spaces.) 4) Suppose that A₁ and A2 are affine subspaces of A, and A₁ A2. Show that dim A₁ ≤ dim A2 with equality holds if and only if A₁ = A2. Hint: pick a point p in A₁ so that one can turn the problem into a problem in linear algebra. Exercise 14. Let A be an affine space, B be an affine subspace of A, and p be a point in B. 1) Show that the pointed space (B, p) is linear subspace of the vector space (A, p). So (B, p) is a vector space as well. 2) Show that B is an affine space modeled on vector space (B, p). (This says that affine subspaces are affine spaces.) 3) Let T: VW be a linear map, w be a vector in W such that T-1 (w) := {ve V | T(v) = w} is a nonempty set. Show that T-1(w) is an affine subspace of the affine space V ( i.e., Vaff), but not a linear subspace of the vector space V unless w = 0. (This says that taking inverse image of a linear map will get us out of the realm of vector spaces.) 4) Suppose that A₁ and A2 are affine subspaces of A, and A₁ A2. Show that dim A₁ ≤ dim A2 with equality holds if and only if A₁ = A2. Hint: pick a point p in A₁ so that one can turn the problem into a problem in linear algebra.