6.1 Dot Products and Orthogonality & 6.2/6.3 Orthogonal Sets, Bases, and Projection

Question 1

best fit line

Answer 1

an appox solution

Question 2

distance of any vector x in Rn should satisfy:

Answer 2

( 1 ) ||x|| >= 0
( 2 ) ||x|| >= 0 <=> x=0
( 3 ) ||cx||=|c| ||x|| for c in R
( 4 ) ||x+y||<=||x||+||y||

Question 3

dot product of x and y

Answer 3

x * y = x(transpose)y

Question 4

magnitude of x

Answer 4

||x|| = rad(x*x)

Question 5

distance between x and y

Answer 5

d(x,y)=||x-y||

Question 6

unit vector u in Rn

Answer 6

a vector of magnitude 1

Question 7

orthogonal

Answer 7

x * y = 0

Question 8

Cauchy Schwartz Inequality

Answer 8

For x, y in Rn

x*y<=||x|| ||y||

Question 9

angle between x,y in Rn x=!0 and y=!0

Answer 9

theta = [ (x*y)/(||x|| ||y||) ]

Question 10

orthogonal complement

Answer 10

Let W subset of Rn be a subspace. The set of all vectors in Rn orthogonal to every vector in W is called orthogonal complement of W

Question 11

Orthogonal Set

Answer 11

The set {v1, v2, …, vp} subset of Rn is an orthogonal set if vi*vj=0 for i=! j

Question 12

theorem:

an orthogonal set of nonzero vectors…………..

Answer 12

is linearly independent

Question 13

orthogonal basis

Answer 13

a basis for subspace W subset of Rn is an orthogonal basis if that basis is an orthogonal set

Question 14

orthonormal basis

Answer 14

an orthogonal basis consisting of unit vectors

Question 15

orthogonal projection of Y onto subspace W

Answer 15

y^