/Solutions

1. Affine transformations

An affine transformation is a function f:ℝ^m→ℝⁿ of the form f(x) = Mx + b where M is an n×m matrix and b is a column vector.

1. Prove or disprove: if f:ℝ^m→ℝⁿ and g:ℝⁿ→ℝ^k are both affine transformations, then (g∘f) is also an affine transformation.

2. Prove or disprove: if f:ℝⁿ→ℝⁿ is an affine transformation and f^-1 exists, then f^-1 is also an affine transformation.

2. Pythagoras goes mod

Let x and y be vectors in (ℤ_p)ⁿ, where p is a prime.

Show that if (x+y)⋅(x+y) = x⋅x + y⋅y, then either x⋅y = 0 (mod p) or p = 2.

3. Convexity

A set of points S in ℝⁿ is convex if, for any x,y∈S, and any 0 ≤ λ ≤ 1, the point λx + (1-λ)y is in S. (Intuitively, this means that the line segment between any two points in S is also in S; visually, S has no dimples or holes.)

1. Prove or disprove: If f:ℝⁿ→ℝ^m is a linear transformation, and S is convex, then f(S) = { f(x) | x∈S } is convex.

2. Prove or disprove: If f:ℝⁿ→ℝ^m is a linear transformation, and f(S) is convex, then S is convex.