| # Exercises Vectors > Exercises ## Subspaces Consider the set S of points such that Show that S is a subspace. Determine its dimension, and find a basis for it. Consider the set in , defined by the equation Show that the set is an affine subspace of dimension . To this end, express it as , where , and are independent vectors. 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 on a hyperplane , where and are given. Determine the angle between the following two vectors: Are these vectors linearly independent? ## Orthogonalization Let 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 : 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 , where 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 , with values Express as a scalar product, that is, find such that for every . Find a basis for the set of points such that . For , we consider the ‘‘power law’’ function , with values Justify the statement: ‘‘the coefficients provide the ratio between the relative error in to a relative error in ’’. Find the gradient of the function that gives the distance from a given point to a point .