Q1: Let $\langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{C}^n$. What is not correct in general?

A1: $\langle x, y \rangle = 0$ implies $x = y$

A2: $\langle x, x \rangle = 0$ implies $x = 0$

A3: $\langle x, 0 \rangle = 0$ for all $x \in \mathbb{C}^n$

A4: $\langle x, i y\rangle = - \langle i x, y\rangle$ for all $x,y \in \mathbb{C}^n$

Q2: Let $\langle \cdot, \cdot \rangle$ be the standard inner product on $\mathbb{C}^n$. What is in general correct?

A1: $\langle x, y \rangle = x^\ast y$

A2: $\langle x, y \rangle = x^T y$

A3: $\langle x, y \rangle = 0$ for all $x,y \in \mathbb{C}^n$

A4: $\langle x, y\rangle = x -y$

Q3: Let $\langle \cdot, \cdot \rangle$ be the inner product on $\mathcal{P}([-1,1], \mathbb{R})$ given by $\langle f, g\rangle = \int_{-1}^1 f(x) g(x), dx$. Which polynomials are orthogonal to each other? Note that $f,g$ are called orthogonal if $\langle f, g\rangle =0$.

A1: $f(x) = x^2$ and $g(x) = x$

A2: $f(x) = x^2$ and $g(x) = 1$

A3: $f(x) = x$ and $g(x) = x$

A4: $f(x) = x^2$ and $g(x) = x^2$