Page 509 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Show that if is a nonzero vector, then the standard matrix for the orthogonal projection of onto the line
9. span is

Let W be the plane with equation .
10.

(a) Find a basis for W.

(b) Use Formula 6 to find the standard matrix for the orthogonal projection onto W.

(c) Use the matrix obtained in (b) to find the orthogonal projection of a point onto W.

(d) Find the distance between the point and the plane W, and check your result using Theorem 3.5.2.

Let W be the line with parametric equations
11.

(a) Find a basis for W.

(b) Use Formula 6 to find the standard matrix for the orthogonal projection onto W.

(c) Use the matrix obtained in (b) to find the orthogonal projection of a point onto W.

(d) Find the distance between the point and the line W.

In , consider the line l given by the equations and the line m given by the equations

12. . Let P be a point on l, and let Q be a point on m. Find the values of t and s that minimize the

distance between the lines by minimizing the squared distance .

For the linear systems in Exercise 3, verify that the error vector resulting from the least squares solution x is
13. orthogonal to the column space of A.

Prove: If A has linearly independent column vectors, and if is consistent, then the least squares solution of

14. and the exact solution of are the same.

Prove: If A has linearly independent column vectors, and if b is orthogonal to the column space of A, then the least

15. squares solution of is .