8 EXERCISES

8.1. Subspaces

1. Consider the set {\bf S} of points such that

    \begin{align*}x_1 + 2x_2 + 3x_3 &= 0, \\ 3x_1 + 2x_2 + x_3 &= 0.\end{align*}

    Show that {\bf S} is a subspace. Determine its dimension, and find a basis for it.\\

2. Consider the set in \mathbb{R}^3, defined by the equation

    \begin{align*}P:=\{x \in \mathbb{R}^3: x_1 + 2x_2 +3x_3 = 1\}.\end{align*}

a. Show that the set P is an affine subspace of dimension 2. To this end, express it as

    \begin{align*}x^0 + {\bf span}(x^1, x^2)\end{align*}

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-Schwarz inequality, prove that the minimum-distance point is proportional to a:=(1, 2, 3).)

8.2. Projections, scalar product, angles

1. Find the projection z of the vector x &= \begin{pmatrix} 2\\ 1 \end{pmatrix} on the line that passes through x_0 &= \begin{pmatrix} 1\\ 2 \end{pmatrix} with direction given by the vector u &= \begin{pmatrix} 1\\ 1 \end{pmatrix}.

2. Find the Euclidean projection of a point x^0 \in \mathbb{R}^n on a hyperplane

    \begin{align*} {\bf P} =\{x: a^Tx= b\}, \end{align*}

where a \in \mathbb{R}^n and b \in \mathbb{R} are given.\\

3. Determine the angle between the following two vectors:

    \begin{align*}x &= \begin{pmatrix} 1\\ 2 \\ 3 \end{pmatrix},\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! & y &= \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}\end{align*}

Are these vectors linearly independent?

8.3. Orthogonalization

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.

8.4. Generalized Cauchy-Schwarz inequalities

1. Show that the following inequalities hold for any vector x:

    \begin{align*}||x||_\infty \leq ||x||_1 \leq n||x||_\infty.\end{align*}

2. Show that following inequalities hold for any vector:

    \begin{align*}||x||_2 \leq ||x||_1 \leq \sqrt{n} ||x||_2\end{align*}

Hint: use the Cauchy-Schwarz inequality for the second inequality.

3. In a generalized version of the above inequalities, show that for any non-zero vector x,

    \begin{align*}1 \leq {\bf Card}(x) \leq \frac{||x||_1^2}{||x||_2^2}\end{align*}

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?

8.5. Linear functions

1. 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

    \begin{align*}f(x) = x_m - \frac{1}{n} \sum\limits_{i=1}^n x_i.\end{align*}

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.\\

2. For \alpha \in \mathbb{R}^2, we consider the ‘‘power-law’’ function f: \mathbb{R}_{++}^2 \rightarrow \mathbb{R}, with values

    \begin{align*}f(x) = x_1^{\alpha_1} x_2^{\alpha_2}.\end{align*}

Justify the statement: ‘‘the coefficients \alpha_i provide the ratio between the relative error in f to a relative error in x_i’’.\\

3. 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.

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