# 1. Affine transformations

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

Prove or disprove: if f:ℝ

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

^{n}→ℝ^{n}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 ℝ^{n} 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.)

Prove or disprove: If f:ℝ

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

^{n}→ℝ^{m}is a linear transformation, and f(S) is convex, then S is convex.