6.1 Inner Products Hwk: (1-4)b, 5a, (6-12)b, 13a, 16b, 17a, 20, 22, 24, 27b, 28abc, 31, 32, 34
This generalizes the dot product, now called the "Euclidean inner product"
Any operation which satisfies these properties of the "dot product" is defined as an "inner product":
1. <u,v> = <v,u> "Symmetry" (commutative)
2. <u+v,w> = <<u,w> + <v,w> "Additivity" (distributive)
3. <ku,v> = k<u,v> "Homogeneity" (scalar mult is associative)
4. <v,v> ³ 0, and <v,v> = 0 iff v = 0 "Positivity"
The examples in the text are specific inner products you should be familiar with:
Ex 1. Euclidean inner product on Rn
Ex 2. Weighted Euclidean inner product Application: mean of a frequency dist
Ex 3: Norm and distance: (these yield the ordinary norm and dist in Rn with the Euclidean inner product)
norm = <u,u>^(1/2)
dist(u,v) = norm(u-v)
Ex 4: shows how these change when using a weighted Euclidean inner product
Unit circle or unit sphere:
set of points in V that satisfy norm(u) = 1
Ex 5 shows a "unit circle" with a weighted inner product: result is an ellipse
For different inner products, distortions occur, but many results of ordinary geometry still apply (such as triangle inequality)
More specific inner product spaces:
Before Ex 6: inner product generated by a matrix A: <u,v> = Au · Av
Alternate forms that use only matrix multiplication, not the inner product notation:
<u,v> = (Av)TAu
= vTATAu
Ex 6: The identity matrix generates the ordinary Euclidean dot product
After Ex 6: The weighted Euclidean inner product is generated by a diagonal matrix whose entries are the square roots of the weights.
Ex 7: For any 2´2 matrices, the sum of the product of corresponding elements is an inner product:
<U,V> = tr(UTV) = tr(VTU) = u1v1 + u2v2 + u3v3 + u4v4
Ex 8: On P2, <p,q> = sum of products of coefficients of like terms
Text shows what the norm and the unit circle look like
Ex 9 (calculus) For f,g in C[a,b], define <f,g> = int(f(x)·g(x),x=a..b)
Text proves axioms hold
Ex 10 describes the norm and unit sphere for this space
Norm: int(f2(x),x=a..b)
Unit sphere: the set of all functions for which norm = 1, i.e. int(f2(x),x=a..b) = 1
Caution: the norm in this inner product space is not the same as the arc length
Thm 1: Basic algebraic properties of inner products
These are extensions of the axioms, and match basic properties of the dot product
Ex 11: Shows how FOIL works for the inner product in any inner product space
6.2 Angle and Orthogonality in Inner Product Spaces Hwk: 1b, 2, 3, 4, 5b, 6b, 7, 8b, 10b, 13b, 15b, 17, 18b, 19, 21, 23, 28, 30, 34
Thm 1: Cauchy-Schwarz Inequality
See Proof
Thm 2 and 3: Properties of length and distance. Notice positivity, homogeneity (for length), symmetry (for distance), and triangle inequality (for both)
Use Cauchy-Schwarz to prove triangle inequality
Thm 4: Generalized Pythagorean Thm
Proof follows from <u,v> = 0
Ex: show x and x3 are orthogonal, and demonstrate
Cauchy-Schwarz
Triangle Ineq
Pythagorean Thm
Def: Orthogonal complement: Let W be subspace of vector space V
1. A vector u in V is orthogonal to W if it is orthogonal to every vector in W
2. The set of all such vectors in V is the orthogonal complement of W
3. We write W┴
Thm 5: Properties of W┴
1. It's a subspace
2. The only vector common to W and W┴ is 0
3. (W┴ )┴ = W
Thm 6: If A is mxn matrix
1. Nullspace(A) ┴ rowspace(A) in Rn
2. Nullspace(AT) ┴ colspace(A) in Rm
Ex 6: To find a basis for the orthogonal complement of a set of vectors
1. Let A be the matrix with the given row vectors
2. Find the basis for the nullspace (solve Ax=0)
3. Express the result as column vectors.
Thm 7 adds Thm 6 to the "big thm"
6.3
Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition Hwk:
1ab, 2ab, 4ab, 5a, 6b, 7b, 10b, 11b, 12, 15b, 17a, 18, 20, 21, 29 ,30 [omit
QR decomposition]
Contrast thm 1 with thm 5a. Let W be a subset of vectors in a vector
space V.
In thm 1, u is in span(W), so
we get coefficients that give u itself
In thm 5, u is in general not
in span W, so we get the closest approx to u that is possible in span(V), i.e.
projwu
So, to find vector orthog to u, take v = u –
projwu. Adding v / norm(v) to
the set W, we get a new set V that includes the new vector in its span and is
still orthonormal.
Good example:
X19. Calculate q1 and q2 by Gram
Schmidt. Then, when you calculate projw2u3,
the result is u3 again, showing that u3 is in span(W2). Hence the orthonormal basis is {q1,q2}
6.4 Best Approximation; Least Squares Hwk: 1a, 2b, 3b, 4b, 5b, 6, 8b, 10, 13, (see 14)
Also do lesson 9.3 #2, 3, 4 (check with calculator LinReg, P2Reg, P3Reg)
6.5 Change of Basis Hwk: 1b, 2b, 3b, 4, 6, 8, 10
Change of basis problem: If we change the basis of a vector space V from an old basis B to a new basis B', how is the old coordinate vector [v]B of a vector v related to the new coordinate vector [v]B'
Solution:
Find the coordinate vectors of the new basis vectors in terms of
the old basis vectors. Form a
"transition matrix" P whose columns are these coordinate
vectors. Then [v]B = P [v]B'
It should be clear that the transition matrix
from B' to B will be P–1
6.6 Orthogonal Matrices Hwk: 1ab, 2ab, 3abcd, 5, 6, 19
Be able to prove thms 1 and 2
Return to: Merced College; Don Power Updated 04/25/07 by Don Power