---
title: 18.156 problem set 01
description: attempts and solutions for 18.156 problem set 01
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## 1. Practicing Cauchy-Schwarz and double counting

> \[!question\] problem 1
>
> (Practicing Cauchy-Schwarz and double counting) Suppose that $X, Y$ are finite sets and $f: X \rightarrow Y$. If $|Y| \le (1/2) |X|$ prove that
>
> [[courses/18-156-projection-theory-spring-2025/notes/pset-01#^problem-1|the full statement]]

^problem-1

### attempt

## 2. On your own, work through the proof of Theorem 1 and Theorem 2

> \[!question\] problem 2
>
> On your own, work through the proof of Theorem 1 and Theorem 2. (You don’t have to turn anything in.) The proofs are based on linear algebra and on orthogonality. Here is a sketchy outline.
>
> [[courses/18-156-projection-theory-spring-2025/notes/pset-01#^problem-2|the full statement]]

^problem-2

### attempt

## 3. Suppose that $P \subset \mathbb{F}_q^d$ is an affine $k$-plane

> \[!question\] problem 3
>
> Suppose that $P \subset \mathbb{F}_q^d$ is an affine $k$-plane. Define the perpendicular subspace $P^\perp$ by
>
> [[courses/18-156-projection-theory-spring-2025/notes/pset-01#^problem-3|the full statement]]

^problem-3

### attempt

## 4. Suppose that $X \subset \mathbb{F}_q^3$

> \[!question\] problem 4
>
> Suppose that $X \subset \mathbb{F}_q^3$. Suppose that $D \subset Gr_q(1,3)$. Let $S = S(X,D) = \max_{W \in D} | \pi^W(X)|$.
>
> [[courses/18-156-projection-theory-spring-2025/notes/pset-01#^problem-4|the full statement]]

^problem-4

### attempt