Highlights
Math 304
Linear Algebra
From last time:
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Harold P. Boas
To find the least-squares solution to Ax = b, solve the
related problem AT Ax = AT b.
Today:
boas@tamu.edu
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inner products and norms
June 21, 2006
Inner product = generalization of scalar product
y1
x1
is
and
In
the standard scalar product of vectors
y2
x2
x1 y1 + x2 y2 .
In an anisotropic problem, one might want to use a weighted
scalar product, such as 2x1 y1 + 5x2 y2 . This will change the
notions of length and angle.
Examples of inner products
R2,
In general, an inner product hx, yi must satisfy three properties:
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symmetry: hx, yi = hy, xi
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linearity: hx + y, zi = hx, zi + hy, zi and hcx, yi = chx, yi for
every scalar c
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positivity: hx, xi is positive unless x = 0.
The norm
p associated to an inner product is given by
kxk = hx, xi .
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The standard scalar product on R 3 is the basic example of
an inner product.
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On a vector space of functions, a common inner product is
Rb
integration: hf , gi = a f (x)g(x) dx.
Example. Show that for the inner product corresponding to
integration over the interval [−1, 1], the Legendre polynomials
1, x, and 12 (3x 2 − 1) are orthogonal to each other.
Solution. Show that the inner products are equal to 0.
R1
h1, xi = −1 x dx = 0 by symmetry.
1
R1
h1, 12 (3x 2 − 1)i = −1 21 (3x 2 − 1) dx = 12 x 3 − x −1 = 0.
R1
hx, 12 (3x 2 − 1)i = −1 12 (3x 3 − x) dx = 0 by symmetry.
More examples in function spaces
Normed linear spaces
R1
Example. For the inner product hp, qi = −1 p(x)q(x) dx, find
the norm of the function p(x) = x.
qR
p
p
1
2
Solution. kxk = hx, xi =
2/3.
−1 x dx =
Without an inner product, we cannot talk about angles or
projections or the Pythagorean law or the Cauchy-Schwarz
inequality, but we can still talk about distance if our vector
space has a norm.
A norm must satisfy three properties:
Example. For the same inner product, find the projection of the
function p(x) = 1 on the function q(x) = 3x 2 .
q
q
Solution. We want hp, kqk
i kqk
or
R1
2
−1 3x dx
R1
4
−1 9x dx
hp,qi
hq,qi q,
which equals
2
5
3x 2 = x 2 .
q(x) =
18/5
3
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scaling: kcvk = |c| kvk for every scalar c
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triangle inequality: kv + wk ≤ kvk + kwk
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positivity: kvk is positive unless v = 0
Examples of norms on R 2
q
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the usual Euclidean norm: k(x1 , x2 )T k2 =
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the taxicab norm: k(x1 , x2 )T k1 = |x1 | + |x2 |
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the maximum norm: k(x1 , x2 )T k∞ = max(|x1 |, |x2 |)
x12 + x22