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MATH69.217* ANSWERS TO
TEST 3 Fall 2001
- (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:
- 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 .
- 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
- Find the QR factorization of the matrix
- Write , and for the columns of . Put . Let
Upon normalization, we get an ONB:
The matrix in the -decomposition
and the matrix is
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C.K. Fong 2001-11-09