Mathematics 369
Homework (due Apr 19)
A. Hulpke
61) Give an example of two vectors v, w ∈ R2 such that {v, w} is linearly dependent, but
we do not have that v = αw for any α ∈ R.
(Hint: The definition of linear independence is slightly different!)

1 −2 −1
1 −1 . Then χA (x) = (x − 3)2 (x + 23 ).
62) Let A =  −2
5
−1 −1
2
a) Show that A is diagonalizable.
b) Determine an orthogonal matrix S such that S−1 AS is diagonal.

63) (The spectral theorem in the formulation used in Physics) Let V an inner product space and L:V → V self-adjoint. (You may assume F = R if you want.) Show that
if {b1 , . . . , bn } is an orthonormal basis of V consisting of eigenvectors of L and λi is the
eigenvalue for bi , then for every v ∈ V we have that
n
L(v) = ∑ λi v, bi bi
i=1
(The physicists write this in the form L = ∑i λi | bi i · hbi |.)
64) Let V be an inner product space.
a) Let S ≤ V be a subspace and PS :V → S the orthogonal projection onto S. Show that PS is
self-adjoint and that PS2 = PS .
b) Let P:V → V be a self-adjoint operator such that P2 = P. Show that P must be the
orthogonal projection onto a subspace of V .
65∗ )
6:
We want to find the points (x, y) in the plane that fulfill the equation 8x2 −4xy+11y2 =
8 −2
a) Consider the symmetric matrix A =
. Determine an orthogonal matrix S
−2 11
such that D = ST AS is diagonal.
b) Describe the set of vectors w such that wT Dw = 6.
b) For v = (x, y)T we have that vT Av = 8x2 − 4xy + 11y2 . Using this fact, and the results from
a) and b), describe the set of points that fulfill 8x2 − 4xy + 11y2 = 6.


1 −1
2 .
66) Determine a singular value decomposition of A =  −2
2 −2
Problems marked with a ∗ are bonus problems for extra credit.