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Unformatted text preview: se 6: 16 points (3 points + 13 points) ∞.
0, 1 and 1 with corresponding Find Consider the matrix 3 .
1 2 0.
123 a) Justify that the matrix A has a QR-decomposition.
b) Find the matrices Q and R of this decomposition.
(4 points for each question)
0 for every weighted inner product on
a) Find two distinct non-zero vectors u and v in R2, such that
b) Find two distinct non-zero vectors u and v in R2, such that
0 for any inner product on R2.
is an orthonormal basis of a real inner product space V. Show that for every
c) Suppose that
d) Let u and v be vectors in an inner product space.
if and only if
e) Let A be a square matrix such that
. What can you say about the eigenvalues of A?
f) Show that the matrix √
√ √ is orthogonal. √ Definition: A proper orthogonal matrix is an orthogonal matrix of which the determinant is equal to 1.
Check that P is not proper and deduce from P a proper orthogonal matrix P’. (Use a relevant small change only)...
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