If you find any mistakes, please make a comment! Thank you.

## Check if a form is an inner product

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.3 Solution: Because $A=A^*$, we have$\overline{g(X,Y)}=(Y^*AX)^*=X^*AY=g(Y,X).$It is also clear that $g$ defines a form. Hence we only need to check…

## Solution to Mathematics for Machine Learning Exercise 3.9

Let $n\in\mathbb N$ and let $x_1,\dots,x_n>0$ be $n$ positive real numbers so that $x_1+\dots+x_n=1$. Use the Cauchy-Schwarz inequality and show that a. $\sum_{i=1}^n x_i^2\geqslant \frac{1}{n}$. b. $\sum_{i=1}^n \frac{1}{x_i}\geqslant n^2$. Hint:…

## Solution to Linear Algebra Hoffman & Kunze Chapter 8.1

Exercise 8.1.1 Let $V$ be a vector space and $(\ | \ )$ an inner product on $V$. (a) Show that $(0|\beta)=0$ for all $\beta$ in $V$. (b) Show that…