Chapter 4 (Vector Spaces)

Question 1

When is a set closed under op *

Answer 1

If a * b is in A (for all a and b in A)

Question 2

Axiom 1

Answer 2

u + v is in v (closed under +)

Question 3

Axiom 2

Answer 3

u + v = v + u (commutative)

Question 4

Axiom 3

Answer 4

u + (v + w) = (u + v) + w (associative)

Question 5

Axiom 4

Answer 5

0 + u = u + 0 = u
–> 0 is an object in V (zero vector)

Question 6

Axiom 5

Answer 6

u + (-u) = (-u) + u = 0
–> each u in V has a has a -u in V

Question 7

Axiom 6

Answer 7

ku is in V (closed under scalar multiplication

Question 8

Axiom 7

Answer 8

k(u+v) = ku + kv

Question 9

Axiom 8

Answer 9

(k+m)u = ku + mu

Question 10

Axiom 9

Answer 10

k(mu) = (km)u

Question 11

Axiom 10

Answer 11

1u = u

Question 12

What are the conditions for W to be a subspace of V

Answer 12

If under + and scalar multiplication on V, W is also a vector space
REMEMBER!
- If axioms 2, 3, 7, 8, 9 are valid in V –> also valid in W
- Just have to check 1 and 6
1 –> u + v is in W
6 –> ku is in W

Question 13

How can vector W be a linear combination of vectors v1,v2,…,vr

Answer 13

If w = k1v1 + k2v2, … +…krvr
–> linear combination of v1,v2,…vr

Question 14

What is a space spanned by v1,v2…,vr (Set S)

Answer 14

Subspace W of all linear combinations of v1,v2,…vr
–> W = span(S) = {k1v1 + k2v2 …+krvr)

Question 15

span {v1} and eq

Answer 15

line through origin
–> span{v1} = tv1 (k1v1)
–> line // to v1

Question 16

span {v1, v2} and eq

Answer 16

plane through origin // to v1 and v2
–> span {v1, v2} = {t1v1 + t2v2} (k1v1+ k2v2)

Question 17

What is a linearly independent set (2)

Answer 17

1. If k1v1 + k2v2 +…+ knvn = 0 has only the trivial soln
   –> if k1=0, k2 =0, …, kn =0
   –> spanned space (subspace) has only the trivial soln
2. (v1 NOT = kv2)
   If no other vector is expressed as a linear combination of other vectors

Question 18

What is a linearly dependent set (2)

Answer 18

1. If k1v1 + k2v2 +…+ knvn = 0 has other soln than trivial
   –> space spanned (subspace) has other solutions
2. v1 = kv2
   If at least one vector can be expressed as a linear combination of other vectors

Question 19

Give an example of a linearly dependent and independent set

Answer 19

1. Linearly dependent: Finite set containing a zero vector
2. Linearly independent: Set with 2 vectors that are not scalar multiples of each other