Solution to Mathematics for Machine Learning Exercise 4.1
Compute the determinant using the Laplace expansion (using the first row) and the Sarrus rule for $$A=\begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6\\ 0 & 2…
Solution to Mathematics for Machine Learning Exercise 3.9
Let $n\in\mathbb N$ and let $x_1,\dots,x_n>0$ be $n$ positive real numbers so that $x_1+\dots+x_n=1$. Use the Cauchy-Schwarz inequality and show that a. $\sum_{i=1}^n x_i^2\geqslant \frac{1}{n}$. b. $\sum_{i=1}^n \frac{1}{x_i}\geqslant n^2$. Hint:…
Solution to Mathematics for Machine Learning Exercise 3.4
Compute the angle between $$\mathbf x=\begin{bmatrix} 1\\ 2\end{bmatrix},\quad \mathbf y=\begin{bmatrix} -1\\ -1\end{bmatrix}$$ using a. $\langle \mathbf x,\mathbf y\rangle := \mathbf x^\top \mathbf y$. b. $\langle \mathbf x,\mathbf y\rangle := \mathbf…
Solution to Mathematics for Machine Learning Exercise 3.3
Compute the distance between $$\mathbf x=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad \mathbf y=\begin{bmatrix}-1\\-1\\0\end{bmatrix}$$ using a. $\langle \mathbf x,\mathbf y\rangle :=\mathbf x^\top \mathbf y$ b. $\langle \mathbf x,\mathbf y\rangle :=\mathbf x^\top \mathbf A\mathbf y$, $\mathbf A=\begin{bmatrix}2&1&0\\1&3&-1\\…
Solution to Mathematics for Machine Learning Exercise 3.2
Consider $\mathbb R^2$ with $\langle ,\rangle$ defined for all $\mathbf x$ and $\mathbf y$ in $\mathbb R^2$ as $$\langle \mathbf x,\mathbf y\rangle :=\mathbf x^\top\mathbf A\mathbf y,\quad \mathbf A:=\begin{bmatrix}2 & 0\\…
Solution to Mathematics for Machine Learning Exercise 3.1
Show that $\langle ,\rangle$ defined for all $\mathbf x=[x_1,x_2]^\top\in\mathbb R^2$ and $\mathbf y=[y_1,y_2]^\top\in\mathbb R^2$ by $$\langle \mathbf x,\mathbf y\rangle := x_1y_1-(x_1y_2+x_2y_1)+2(x_2y_2)$$ is an inner product. Solution: We check it by…
Solution to Mathematics for Machine Learning Exercise 2.1
We consider $(\mathbb R\setminus \{-1\},\star)$, where \begin{equation}\label{2.1.1}a\star b:= ab+a+b,\quad a,b\in\mathbb R\setminus \{-1\}\end{equation} a. Show that $(\mathbb R\setminus \{-1\},\star)$ is an Abelian group. b. Solve $$3\star x \star x = 15$$…
Solution to Mathematics for Machine Learning
Chapter 2 Linear Algebra #2.1, #2.2, #2.3, #2.4, #2.5, #2.6, #2.7, #2.8, #2.9, #2.10, #2.11, #2.12, #2.13, #2.14, #2.15, #2.16, #2.17, #2.18, #2.19, #2.20 Chapter 3 Analytic Geometry #3.1, #3.2,…
Solution to Linear Algebra Done Wrong Exercise 1.3.7
Show that any linear transformation in $\mathbb C$ (treated as a complex vector space) is a multiplication by $\alpha\in\mathbb C$. Solution: Let $T$ be a linear transformation from $\mathbb C$…
Solution to Linear Algebra Done Wrong Exercise 1.3.5
Let $A$ be a linear transformation. If ${\bf z}$ is the center of the straight interval $[{\bf x}, {\bf y}]$, show that $A{\bf z}$ is the center of the interval…