Summary linear algebra (notes, exercises, correction)

CHAPTER 6 ORTHOGONALITY

6.1 Inner Product, Length and Orthogonality


• Inner Product

   
u v
 1  1
Let u =  ... , v =  ... . The inner product (dot product) of u and v is
   
   
un vn
defined as
u · v = u1 v1 + . . . + un vn = uT v


   
2 3
   
• Example Let u = −5, v =  2 . Find u · v and v · u.
   
   
−1 −3




• Theorem

• u·v =v·u
• (u + v) · w = u · w + v · w
• (cu) · v = c(u · v) = u · (cv)
• u · u ≥ 0, and u · u = 0 if and only if u = 0.



1

,2

• The Length of a Vector


The length (or norm) of a vector v is
q
kvk = v12 + . . . + vn2 .

It follows that
kvk2 = v · v.




• Example If v is a vector in Rn , show that kcvk = |c|kvk.




 
1
 
def −2
 
• Example Let v = (1, −2, 2, 0) =  . Find a unit vector in the direction
2
 
 
0
of v.




• Distance

The distance between u and v in Rn is

dist (u, v) = ku − vk

, 3

• Cauchy-Schwarz Inequality



|u · v| ≤ kuk kvk




• Angle

If the angle between two vectors u and v is θ, then
u·v
cos θ = .
kuk kvk
Two vectors u and v are orthogonal if and only if

u · v = 0.




• Example Prove that two vectors u and v in Rn are orthogonal if and only if

ku + vk2 = kuk2 + kvk2

, 4

• Orthogonal Complement

Let W be a subspace of Rn . The orthogonal complement of W , denoted by
W ⊥ is
W ⊥ = {z ∈ Rn : z · w = 0 for all w ∈ W }

W ⊥ is a subspace of Rn .




• Theorem

Let A be an m × n matrix.
• The orthogonal complement of the row space of A is the null space of A.
• The orthogonal complement of the column space of A is the null space of
AT .




Exercises §6.1, 2,6,11,14,16,17,23,24,27,31.