---
title: Quiz 4
tags: Linear algebra
GA: G-77TT93X4N1
---

# Quiz 4

* Given $n\in\mathbb{N}$, find a constant $C$ such that, for any ${\bf v}\in\mathbb{R}^n$,
$$
\|{\bf v}\|_1 \le C\|{\bf v}\|_2.
$$
* Find counter-examples for the following statements:
* Let $L:V\to W$ be a linear transformation. Then $\langle w, L(v)\rangle=0$ for all $v\in V$ implies $w=0$.
* Let $L:V\to W$ be linear, $V$ is a vector space and $W$ is an inner product space. Define
$$
\langle u, v \rangle_L := \langle L(u), L(v) \rangle, \quad u,v\in V,
$$
where the right hand side involves the given inner product on $W$. Then the above operation, $\langle\cdot, \cdot \rangle_L$, defines an inner product on $V$.
* Find a polynomial $q(x)\in\mathbb{P}_2(\mathbb{R})$ such that
$$
p(\frac{1}{2})=\int^1_0 p(x)q(x)\,dx,
$$
for any $p(x)\in\mathbb{P}_2(\mathbb{R})$.
* Find a polynomial $q(x)\in\mathbb{P}_2(\mathbb{R})$ such that
$$
\int^1_0 p(x)\cos(\pi x)\,dx=\int^1_0 p(x)q(x)\,dx,
$$
for any $p(x)\in\mathbb{P}_2(\mathbb{R})$.

---

# Quiz 04/17

* [quiz0417](https://drive.google.com/file/d/1criAisAZn6ui7E256VLAY9Xkv8bkgh15/view?usp=sharing)