Exercises

Subspaces

Consider the set ${\bf S}$ of points such that
$x_1 + 2x_2 + 3x_3 = 0, 3x_1 + 2x_2 + x_1 = 0.$
Show that ${\bf S}$ is a subspace. Determine its dimension, and find a basis for it. $\\$
Consider the set in $\mathbb{R}^3$ , defined by the equation
$P:=\{x \in \mathbb{R}^3: x_1 + 2x_2 +3x_3 = 1\}.$
a. Show that the set $P$ is an affine subspace of dimension $2$ . To this end, express it as $x^0 + {\bf span}(x^1, x^2)$ , where $x^0 \in P$ , and $x^1, x^2$ are independent vectors.
b. Find the minimum Euclidean distance from $0$ to the set $P$ . Find a point that achieves the minimum distance. (Hint: using the Cauchy-Schwartz inequality, prove that the minimum-distance point is proportional to $a:=(1, 2, 3)$ .)

Find the projection $z$ of the vector $x=(2, 1)$ on the line that passes through $x_0 = (1,2)$ and with direction given by the vector $u = (1, 1)$ . $\\$
Find the Euclidean projection of a point $x^0 \in \mathbb{R}^n$ on a hyperplane ${\bf P} =\{x: a^Tx= b\}$ , where $a \in \mathbb{R}^n$ and $b \in \mathbb{R}$ are given. $\\$
Determine the angle between the following two vectors:
(1) $\begin{align*} x &= \begin{pmatrix} 1\\ 2 \\ 3 \end{pmatrix} \end{align*},$

(2) $\begin{align*} y &= \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \end{align*}$

Are these vectors linearly independent?

Let $x, y \in \mathbb{R}^n$ be two unit-norm vectors, that is, such that $||x||_2 = ||y||_2 = 1$ . Show that the vectors $x-y$ and $x+y$ are orthogonal. Use this to find an orthogonal basis for the subspace spanned by $x$ and $y$ .

Show that the following inequalities hold for any vector $x$ :
$||x||_\infty \leq ||x||_1 \leq n||x||_\infty.$ [latex]\\[/latex]
Show that following inequalities hold for any vector $x$ :
$||x||_2 \leq ||x||_1 \leq \sqrt{n} ||x||_2$
Hint: use the Cauchy-Schwartz inequality for the second inequality.
In a generalized version of the above inequalities, show that for any non-zero vector $x$ ,
$1 \leq {\bf Card}(x) \leq \frac{||x||_1^2}{||x||_2^2}$
where ${\bf Card}(x)$ is the cardinality of the vector $x$ , defined as the number of non-zero elements in $x$ . For which vectors $x$ is the upper bound attained?

For a $n$ -vector $x$ , with $n=2m-1$ odd, we define the median of $x$ as $x_m$ . Now consider the function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ , with values
$f(x) = x_m - \frac{1}{n} \sum\limits_{i=1}^n x_i.$
Express $f$ as a scalar product, that is, find $a \in \mathbb{R}^n$ such that $f(x) = a^T x$ for every $x$ . Find a basis for the set of points $x$ such that $f(x) = 0$ . $\\$
For $\alpha \in \mathbb{R}^2$ , we consider the ‘‘power-law’’ function $f: \mathbb{R}_{++}^2 \rightarrow \mathbb{R}$ , with values
$f(x) = x_1^{\alpha_1} x_2^{\alpha_2}.$
Justify the statement: ‘‘the coefficients $\alpha_i$ provide the ratio between the relative error in $f$ to a relative error in $x_i$ ’’. $\\$
Find the gradient of the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ that gives the distance from a given point $p \in \mathbb{R}^2$ to a point $x \in \mathbb{R}^2$ .