Exercises

Vectors > Exercises

Subspaces

Consider the set S of points such that

x_1 + 2 x_2 + 3x_3 = 0, ;; 3x_1 + 2x_2 + x_3 = 0.

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

Consider the set in $mathbf{R}^3$ , defined by the equation

$P := left{ x in mathbf{R}^3 ~:~ x_1 + 2x_2 + 3x_3 = 1 right}.$

1. Show that the set is an affine subspace of dimension . To this end, express it as $x^0 + mbox{bf span}(x^1,x^2)$ , where $x^0 in {cal P}$ , and are independent vectors.
2. Find the minimum Euclidean distance from to the set . Find a point that achieves the minimum distance. (Hint: using the Cauchy-Schwartz inequality, prove that the minimum-distance point is proportional to .)

Projections, scalar product, angles

Find the projection of the vector on the line that passes through and with direction given by the vector .
Find the Euclidean projection of a point $x_0 in mathbf{R}^n$ on a hyperplane $mathbf{P} = { x ::: a^Tx = b}$ , where $a in mathbf{R}^n$ and are given.
Determine the angle between the following two vectors:

x = left(begin{array}{c} 1 2 3 end{array}right), ;; y = left(begin{array}{c} 3 2 1 end{array}right).

Are these vectors linearly independent?

Orthogonalization

Let $x,yinmathbf{R}^n$ be two unit-norm vectors, that is, such that . Show that the vectors and are orthogonal. Use this to find an orthogonal basis for the subspace spanned by and .

Generalized Cauchy-Schwartz inequalities

Show that the following inequalities hold for any vector :

Show that following inequalities hold for any vector :

$|x|_2 le |x|_1 le sqrt{n} |x|_2.$

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 ,

$1 le mbox{bf Card}(x) le frac{|x|_1^2}{|x|_2^2},$

where mbox{bf Card}(x) is the cardinality of the vector , defined as the number of non-zero elements in . For which vectors is the upper bound attained?

Linear functions

For a -vector , with odd, we define the median of as . Now consider the function $f : mathbf{R}^n rightarrow mathbf{R}$ , with values

$f(x) = x_m - frac{1}{n} sum_{i=1}^n x_i.$

Express as a scalar product, that is, find $a in mathbf{R}^n$ such that f(x) = a^Tx for every . Find a basis for the set of points such that f(x) = 0 .

For $alpha in mathbf{R}^2$ , we consider the ‘‘power law’’ function $f : mathbf{R}^2_{++} rightarrow mathbf{R}$ , 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 to a relative error in x_i ’’.

Find the gradient of the function $f : mathbf{R}^2 rightarrow mathbf{R}$ that gives the distance from a given point $p in mathbf{R}^2$ to a point $x in mathbf{R}^2$ .