MATH69.217* ANSWERS TO TEST 3 Fall 2001

1.
(a) State the definition of an orthonormal basis in . (b) Verify the identity

where stands for the dot product of vectors and in .

(a) An orthonormal basis in is a basis in consisting of unit vectors perpendicular to each other. (b) We begin with the right hand side:

Done.

2.
Find the entries , and in the following orthogonal matrix

From the fact that the first two columns are perpendicular, we obtain . From the fact that the first and the last rows are perpendicular, we get . Finally, from the fact last two rows are perpendicular, we get .

3.
Find the projection of the vector onto the subspace spanned by the vectors and , where

Begin by checking that is an orthogonal system.

First we check . Hence is indeed and orthogonal system. The required projection is

4.
Find the QR factorization of the matrix

Write , and for the columns of . Put . Let

and

Upon normalization, we get an ONB:

The matrix in the -decomposition of is

and the matrix is