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

# Quiz 1

> Exercises:
* p.5 - 1.1 ~ 1.8
* p.11 - 2.2 ~ 2.6
* p.17 - 3.2, 3.3, 3.6

## Suggested solutions
* 1.2( b) Not a vector space.
* Take $f_1(x) = 1$, then $-1\cdot f_1(x) = -1$ is not a non-negative function. So the set is not closed under scaler multiplication, and is not a vector space.
* 1.2( c) Not a vector space.
* Let $n\ge 1$ and take $p_1(x) = x^n$, and $p_2(x) = -x^n$. Both $p_1$ and $p_2$ are polynomials of degree exactly $n$. But $p_1 + p_2 = 0$ is a polynomial of zero degree. So the set is not closed under vector addition, and is not a vector space.
* 1.3( a) Yes. By definition of a vector space, it contains a zero vector.
* 1.3( b) No. See [Chapter 1 extra note 1](https://hackmd.io/@teshenglin/2024LA2_ch1_1): Theorem 10.
* 1.3( d) No. See a conterexample in 1.2( c).
* Def: A polynomial of degree $n$ is a polynomial with its higest degree of non-zero coefficient exactly equals to $n$.
* 2.2
* 2.3 There are $3$ elements in the basis.

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# Quiz 03/06

* [quiz0306](https://drive.google.com/file/d/1tqZ1QgyyT49J_JdLISJiTMJKyYN0NVrp/view?usp=sharing)