00:00 Introduction

00:20 Definitions

01:58 Remark

03:43 Visualisations

04:58 Credits

Q1: Let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space with $x,y \in X$. What is not correct in general?

A1: If $x = 0$, then $x \perp y$.

A2: If $x \perp y$, then $x$ and $y$ are orthogonal.

A3: If $x \perp x$, then $x = 0$.

A4: If $x - y \perp x$, then $x = 0$.

Q2: Let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space with $x \in X$ and $U, V \subseteq X$. What is not correct in general?

A1: If $x = 0$, then ${x}^\perp = X$.

A2: If $x \perp y$ for all $y \in U$, then ${x} = U^\perp$.

A3: If $U \subseteq V$ then $U^\perp \supseteq V^\perp$.

A4: $U^\perp$ is always a subspace in $X$.

Q3: Let $(\mathbb{C}^n, \langle \cdot, \cdot \rangle)$ be the inner product space with the standard inner product. Are $\binom{i}{1}$ and $\binom{-i}{1}$ orthogonal?

A1: Yes!

A2: No!