§5.4
29. Consider the vector space Rn with inner product < x, y >= xT y. For any
n × n matrix A, show that
a)< Ax, y >=< x, AT y >
b)< AT Ax, x >= kAxk2
Solutions: (Jeff)
a) Using properties from the book we have:
< Ax, y >= (Ax)T y (by the given)
= (xT AT )y (transpose property)
= xT (AT y) (associative property)
=< x, AT y >. (by the given)
b) Using the properties form the book we have:
< AT Ax, x >= (AT Ax)T x (by the given)
= (xT (AT A)T )x (transpose property)
= (xT (AT (AT )T ))x (transpose property)
= (xT AT A)x (double transpose property)
= ((Ax)T A)x (transpose property)
= (Ax)T (Ax) (associative property)
=< Ax, Ax > (by the given)
= kAxk2 . (by the findings of section 5.1)
§5.5
30. Find the best least squares approximation to f (x) = |x| on [−π, π] by a
trigonometric polynomial of degree less than or equal to 2.
Solution: (Joe)
According to Leon, the formula for a trigonometric approximation of degree 2
for a function is given by:
2
f2 (x) =
a0 X
+
(ak cos kx + bk sin kx)
2
k=1
Also, we are shown that the following are the formulas for the coefficients of our
degree 2 approximation:
Z
1 π
a0 =< |x|, 1 >=
|x| dx
π −π
1
1
π
Z
1
π
Z
1
a2 =< |x|, cos 2x >=
π
Z
1
π
Z
a1 =< |x|, cos x >=
b1 =< |x|, sin x >=
b2 =< |x|, sin 2x >=
π
|x| cos x dx
−π
π
|x| sin x dx
−π
π
|x| cos 2x dx
−π
π
|x| sin 2x dx
−π
We know that b1 and b2 are odd functions, so over the interval [−π, π] these will
both equal 0. This leaves us to only need to compute a0 , a1 , and a2 . Since the
integrals defined by the inner products in a0 , a1 , and a2 are symmetric over the
y-axis, we need only to take the following integrals over the interval [0, π] and
then multiply them by two:
π
Z
2 x2 2 π
x dx =
=π
a0 =
π 0
π 2 x=0
Z π
Z
2
4
2
2 π
π
π
sin x dx] = cos x|x=0 = −
x cos x dx = [ x sin x|x=0 −
a1 =
π 0
π
π
π
0
π
π
Z π
Z
2 x sin x sin 2x
2 cos2x 2 π
x cos 2x dx = [
−
=0
dx] =
a2 =
π 0
π
2 x=0
2
π
4 x=0
0
Now, using a0 , a1 , and a2 in the aforementioned formula, we obtain
f2 (x) =
π
4
− cos x
2
π
as our desired degree 2 trigonometric approximation of f (x) = |x|.
2