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

where stands for the dot product of vectors and in . **Answer:**- (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

**Answer:**- 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. **Answer:**- First we check . Hence is indeed and orthogonal system. The required
projection is

- 4.
- Find the QR factorization of the matrix

**Answer:**- 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