Summary of Section 4.6 (Orthogonality)

1. 4. Vector Spaces
ORTHOGONAL
VECTORS
in Rn

2. Defn - Let and be vectors in R2.
The inner product or dot product of u and v,
denoted by ( u,v ) or u•v, is
u•v = u1v1 + u2v2.
Similarly, for and in R3,
u•v = u1v1 + u2v2 + u3v3
Inner (Dot, Scalar) Product in R2 and R3
1 1
2 2
u v
u v
u v
1 1
2 2
3 3
u v
u v
u v
u v

3. Defn - Let V be a real vector space. An inner product
on V is a mapping from V x V to R that assigns, to
each ordered pair of vectors u and v of V, a real
number (u,v) satisfying
a) (u,u) > 0 for u ≠ 0, and (u,u) = 0 if and only if
b) (u,v) = (v,u) for all u, v in V
c) (u+v,w) = (u,w) + (v,w) for all u, v, w in V
d) (cu,v) = c(u,v) for all u, v in V and for all real c
Inner Product for a general Vector Space V

4. The standard inner product on Rn can be defined as
(u,v) = u1v1 + u2v2 + + unvn where
Example: Standard Inner Product in Rn
1 1
2 2
n n
u v
u v
u v
u v

5. Let and be vectors in R2.
Define
(u,v) = u1v1 u2v1 u1v2 + 3u2v2
This gives an inner product on R2.
One can verify that the 4 conditions in the definition
of inner product are satisfied.
Example: Another Inner Product in R2
1 1
2 2
u v
u v
u v

6. Let V be the vector space C[0,1] of all continuous
functions on the interval [0,1].
For f,g Î∈V, the following expression defines an inner
product
Example: Inner Product in C[0,1]
1
0
,f g f t g t dt

7. Defn - A real vector space that has an inner product
defined on it is called an inner product space.
Defn - In an inner product space, the norm (length)
of a vector u is defined as
Since (0,0) = 0, we have
If x is a nonzero vector, then the vector
is a unit vector
Norm (Length, Magnitude)
,u u u
00
u
1
x
x

8. If u and v are any two vectors in an inner product
space V, then
=======================================
If u ≠ 0 and v ≠ 0, then
qθ is called the angle between u and v but it usually
does not have a geometric interpretation
Cauchy-Schwarz Inequality
2 2 2
,u v u v
2
2 2
, ,
1 or 1 1
u v u v
u vu v
,
cos 0
u v
u v

9. Let u and v be vectors in an inner product space V.
Then
Triangle Inequality
u v u v

10. Let V be an inner product space and
let u Î∈ V and v Î∈ V.
Then,
u and v are orthogonal if and only if
(u,v) = 0
Orthogonal Vectors

11. Let and be vectors in R2.
The distance between u and v is defined as the
magnitude of u v, i.e. as
Example R2
1
2
v
v
v
2 2
1 2 ,v v v vv vv
1
2
u
u
u
2 2
1 1 2 2u v u vu v

12. Example R3
1 1
2 2
3 3
u v
u v
u v
u v
2 2 2
1 2 3
22 2
1 1 2 2 3 3
,u u u uu u u
u v u v u v
u
u v

13. Example R3 : Angle
The angle between two vectors 1 1
2 2
3 3
u v
u v
u v
u v
Lengths of sides are , and
The cosine of the angle between two vectors u and v is
v u v u
•cos u v
u v

14. Example : norm
In R3, let and
Using the standard inner product in R3
1 3
2 2
3 2
u v
2 2 2
2 2 2
, 5 14 17
, 25 14 17
,
u v u v
u v u v
u v u v

15. Example : orthogonal
Let V = P2, the set of all polynomials of degree
≤ 2. For p(t), q(t) Î∈ P2 define an inner product
then the vectors t and t -− 2/3 are orthogonal
since
1
0
,p t q t p t q t dt
1 1
2
0 0
, 2 3 2 3 2 3 0t t t t dt t t dt

16. Orthogonal Sets, Spaces
Defn - Let V be an inner product space.
A set S of vectors in V is called orthogonal if every two
distinct vectors in S are orthogonal.
Defn - Let W be a subspace of an inner product
space V. A vector u in V is orthogonal to W if it is
orthogonal to every vector in W
Defn - Let W be a subspace of an inner product
space V. The set of all vectors in V that are
orthogonal to W is called the orthogonal complement
of W in V and is denoted by W^⊥

17. Orthogonal Spaces
Theorem - Let W be a subspace of an inner product
space V. Then
a) W^⊥ is a subspace of V
b) W W^⊥ = { 0 }
For the system Ax = 0
Row(A) and Null(A)
are orthogonal spaces.

18. Orthogonal and Linear Independence
Theorem - Let S = { u1, u2, , un } be a
finite orthogonal set of nonzero vectors in an
inner product space V.
Then S is linearly independent.

19. The End