Ch . 1 Vectors in R^n

Question 1

Axioms for a real vector space

Answer 1

Closure under addition and scalar multiplication
Associativity
Existence of an identity
Existence of an inverse
Distributivity

Question 2

State the Cauchy Schwarz inequality

Answer 2

|v · w| ≤ |v||w|

with equality iff v and w are collinear

Question 3

Finding the angle between 2 vectors

Answer 3

cos(θ) = (v · w) / |v||w|

Question 4

If 2 vectors are orthogonal…

Answer 4

v · w = 0

Question 5

Properties of the vector product

Answer 5

(i) Antisymmetry:
u x v = -v x u

(ii) Linearity in the left factor:
(u + v) x w = u x w + v x w
(λu) x v = λ(u x v)

(iii) Linearity in the right factor:
u x (v + w) = u x w + u x v
u x (λv) = λ( u x v)

(iv) Orthogonality to the input:
v · (v x w) = w · (v x w) = 0

Question 6

What are the closure axioms?

Answer 6

(i) If v and w are in R^n, then v + w is in R^n

(ii) If v is in R^n, the λv is in R^n

Question 7

What are the axioms for addition?

Answer 7

(i) Associativity

(ii) Existence of an additive identity:
There is a vector 0 in R^n such that
v + 0 = v = 0 + v for any v in R^n

(iii) Existence of an additive inverse:
There is an inverse element in R^n -v such that
v + (-v) = 0 = (-v) + v for any v in R^n

(iv) Commutativity

Question 8

What are the axioms for scalar multiplication?

Answer 8

(i) Associativity
(ii) 0v = 0
(iii) Iv = v
(iv) Distributivity

Question 9

Properties of the scalar product

Answer 9

(i) Symmetry:
u · v = v · u

(ii) Linearity in the left factor:
(u + v) · w = u · w + v · w
(λu) · v = λ(u · v)

(iii) Linearity in the right factor:
u · (v + w) = u · w + u · v
u · (λv) = λ(u · v)

(iv) Positivity:
v · v ≥ 0 (with v · v = 0 iff v = 0)

Question 10

How to find the length of the vector product |v x w|

Answer 10

|v x w| = |v||w|sin(θ)

Question 11

What are the 2 ways of describing a plane?

Answer 11

1) Parametric form
π = {x = a + λd1 + μd2 | λ, μ in the reals}

2) Using a normal vector
π = {x in R^3 | n · x = n · a}
Where a is a point on π

3’) Multiplying out 2)
ax + by + cz = l
Where l = n · a

Question 12

What are the 2 ways of describing a line?

Answer 12

1) Parametric form
L = {x = a + λd | λ in the reals}
Where a is a point on L, and d is the direction vector}

2) 2 normal vectors n1, n2 and a point a on L
L = {x in R^3 | n1 · x = n1 · a, n2 · x = n2 · a}

2') Multiplying out 2
n1 = (a,b,c), n2 = (d,e,f)
ax + by + cz = l
dx + ey + fz = m
Where l = n1 · a, m = n2 · a

3) Another description
(x - a1) / d1 = (y - a2) / d2 = (z - a3) / d3

Question 13

Associativity for addition and scalar multiplication

Answer 13

Addition: (u + v) + w = u + (v + w)

Scalar multiplication: λ(μv) = (λμ)v

Question 14

Distributivity for scalar multiplication

Answer 14

(λ + μ)v = λv + μv

λ(v + w) = λv + λw

Question 15

Commutativity for addition

Answer 15

v + w = w + v