Page 532 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Chapter 6

Supplementary Exercises

Let have the Euclidean inner product.
1.

(a) Find a vector in that is orthogonal to and and makes equal angles with
and .

(b) Find a vector of length 1 that is orthogonal to and above and such that the cosine of the

angle between x and is twice the cosine of the angle between x and .

Show that if x is a nonzero column vector in , then the matrix
2.

is both orthogonal and symmetric.

Let be a system of m equations in n unknowns. Show that
3.

is a solution of the system if and only if the vector is orthogonal to every row vector of A in the
Euclidean inner product on .

Use the Cauchy–Schwarz inequality to show that if , , …, are positive real numbers, then
4.

Show that if x and y are vectors in an inner product space and c is any scalar, then
5.

Let have the Euclidean inner product. Find two vectors of length 1 that are orthogonal to all three of the vectors

6. , and .

Find a weighted Euclidean inner product on such that the vectors
7.