---
title: Assignments 3
tags: Linear algebra
GA: G-77TT93X4N1
---

# Assignments 3

1. Exercise 1.5 at p.124.
2. Suppose $V$ is a vector space, $W$ is an inner product space and $T:V\to W$ is linear and injective. For $v_1, v_2\in V$, define
$$
\langle v_1, v_2\rangle:=\langle Tv_1, Tv_2\rangle,
$$
where the right-hand side involves the given inner product on $W$. Prove that this defines an inner product on $V$.
3. Show that
$$
\left(a_1b_1+a_2b_2+\cdots+a_nb_n\right)^2\leq \left(a^2_1+2a^2_2+\cdots+na^2_n\right)
\left(b^2_1+\frac{b^2_2}{2}+\cdots+\frac{b^2_n}{n}\right),
$$
for all $a_1,\cdots,a_n$, $b_1,\cdots,b_n\in\mathbb{R}$, $n\in\mathbb{N}$.
4. Exercise 1.8( c) at p.125.
5. (The problem number is the last digit of your student ID number)
* Use ChatGPT to find the **correct answer** for your assigned problem.
* [Problems](https://drive.google.com/file/d/1mlckS9VNyJA7RoZJdcJFb5wPV4XrYg6h/view?usp=sharing).

# TA solution

* [Assignment3_solution](https://drive.google.com/file/d/1BjWiGQnCRXk8srua2K8FbUroGex1QsNt/view?usp=sharing)