---
title: Assignments 1
tags: Linear algebra
GA: G-77TT93X4N1
---

# Assignments 1

1. Let $V$ be a vector space over a field $F$ with vector addition $\oplus$ and scalar multiplication $\odot$. Assume ${\bf u, v, w}\in V$, solve the following system of equations: **(You must indicate at each step which properties are applied.)**
$$
\begin{align}
& {\bf u} \oplus {\bf v} \oplus (2\odot{\bf w}) = {\bf 0}, \\
& (2\odot{\bf u}) \oplus (2\odot{\bf v}) \oplus {\bf w} = {\bf 0}, \\
& (3\odot{\bf u}) \oplus (4\odot{\bf v}) \oplus (2\odot{\bf w}) = {\bf 0}.
\end{align}
$$
2. Let $V$ be the vector space defined as
$$
V = \{(x_1, x_2, x_3) | x_1, x_2, x_3\in\mathbb{R}^+, x_1+x_2+x_3=1\}, \quad F=\mathbb{R},
$$
with $\oplus$ and $\odot$ defined as
$$
\begin{align}
(x_1, x_2, x_3) \oplus (y_1, y_2, y_3) &= \frac{1}{x_1y_1+x_2y_2+x_3y_3}(x_1y_1, x_2y_2, x_3y_3),\\
\alpha \odot (x_1, x_2, x_3) &= \frac{1}{x_1^{\alpha}+x_2^{\alpha}+x_3^{\alpha}}(x_1^{\alpha}, x_2^{\alpha}, x_3^{\alpha}).
\end{align}
$$
Let ${\bf u}= (\frac{1}{2}, \frac{1}{4}, \frac{1}{4})$, ${\bf v}= (\frac{1}{4}, \frac{1}{2}, \frac{1}{4})$ and ${\bf w}= (\frac{1}{4}, \frac{1}{4}, \frac{1}{2})$, find $\alpha$ and $\beta\in\mathbb{R}$ such ${\bf w} = (\alpha\odot{\bf u}) \oplus (\beta\odot{\bf v})$.
6. Let $V=\mathbb{R}^2$ with $\oplus$ and $\odot$ defined as
$$
(a_1, a_2)\oplus (b_1, b_2) = (a_1+b_1, a_2b_2), \quad \alpha\odot (a_1, b_1) = (\alpha a_1, b_1).
$$
Show that $V$ is not a vector space.
7. Exercise 2.6 at p.11.
8. Exercise 3.3 at p.17.

# TA solution

* [Assignment1_solution](https://drive.google.com/file/d/1unIJQ13izkErjC74QfNpLmpBE81vtwp2/view?usp=sharing)