Chapter 3.2 Flashcards

0
Q

If v=(v(1), v(2), … , v(n)) is a vector in R^(n), what is the norm of v?

A

||v|| = sqrt(v(1)^2, v(2)^2, … , v(n)^2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
1
Q

||v|| can be called three things?

A

1) Norm of v (common mathematical synonym for length)
2) Length of v
3) Magnitude of v

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

If v is a vector in R^(n) and k is any scalar, then (3) (norm)

A

1) ||v|| => 0
2) ||v|| = 0 iff v=0
3) ||kv|| = |k|||v||

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Unit vectors

A

A vector of norm 1. You can obtain a unit vector with the same direction as any vector v by the formula
u = 1/||v||*v This is called “normalizing v”

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Standard unit vectors

A

Unit vectors in positive directions of coordinate axes

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

If the standard unit vectors in R^(n) are e1, e2, … en, then any vector v in R^(n) can be expressed as

A

v = (v(1)e1, v(2)e2, … , v(n)*en)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) then we denote the distance between u and v to be

A

d(u,v) = sqrt((v(1)-u(1))^2, (v(2)-u(2))^2, … , (v(n)-u(n))^2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

The angle v between u and w satisfies

A

0 <= pi

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

If u and w are nonzero vectors in R^(n), and if v is the angle between u and w, then the dot product (also called?) of u and w is

A

u·w = ||u||||w||cos(v)
Also called Euclidian Inner Product
If u=0 or w=0 we define u·w to be 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

How is the angle v expressed through the dot product and what can be said about the angle through examination of the dot product

A

cos(v) = u·w/(||u||*||v||)

1) v is obtuse if u·w < 0
2) v is acute if u·w > 0
3) v is pi/2 uf u·w = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) are vectors in R^(n) then the dot product (also called?) of u and v is denoted u·v and is defined by

A

u·v = (u(1)v(1), u(2)v(2), … , u(n)*v(n))

Also called the Euclidian Inner Product

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

If u, v and w are vectors in R^(n), what correlations between the dot product and the norm is there (5)

A

1) ||v||=sqrt(v·v)
2) v·v = (||v||)²
3) (u·v) = (||u+v||)²
4) |u·v| <= ||u||||v||
if u and v are codirectional:
5) u·v = ||u||
||v||

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

If u, v and w are vectors in R^(n) and k and m are scalars then (5) (dot product)

A

1) u·v = v·u
2) u·(v+w) = u·v + u·w
3) k(u·v) = (ku)·v
4) u·u >= 0 and u·u = o iff u = o
5) 0·v = v·0 = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Cauchy-Schwartz Inequality

A

If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) are vectors in R^(n) then

|u·v| <= ||u||*||v||

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Triangle Inequalities for vectors and distances

A

1) ||u + v|| <= d(u,v) + d(v,w) (for distances)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Parallellogram Equation for vectors

A

If u and v are vectors in R^(n) then

||u+v||² + ||u-v||² = 2(||u||² + ||v||²)

17
Q

The dot product expressed in terms of norms

A

u·v = 1/4||u+v||² + 1/4||u-v||²

18
Q

How is Au·v and u·Av written if you translate A?

A

Au·v = u·A^(T)*v

And u·Av = A^(T)u·v