8 EXERCISES
- Subspaces
- Projections, scalar products, angles
- Orthogonalization
- Generalized Cauchy-Schwarz inequalities
- Linear functions
8.1. Subspaces
1. Consider the set of points such that
Show that is a subspace. Determine its dimension, and find a basis for it.
2. Consider the set in , defined by the equation
a. Show that the set is an affine subspace of dimension . To this end, express it as
where , and are independent vectors.
b. Find the minimum Euclidean distance from to the set . Find a point that achieves the minimum distance. (Hint: using the Cauchy-Schwarz inequality, prove that the minimum-distance point is proportional to .)
8.2. Projections, scalar product, angles
1. Find the projection of the vector on the line that passes through with direction given by the vector
2. Find the Euclidean projection of a point on a hyperplane
where and are given.
3. Determine the angle between the following two vectors:
Are these vectors linearly independent?
8.3. 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 .
8.4. Generalized Cauchy-Schwarz inequalities
1. Show that the following inequalities hold for any vector :
2. Show that following inequalities hold for any vector:
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 ,
where is the cardinality of the vector , defined as the number of non-zero elements in For which vectors is the upper bound attained?
8.5. Linear functions
1. 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 .
2. 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 ’’.
3. Find the gradient of the function that gives the distance from a given point to a point .