Sums And Direct Sums

Question 1

What is sum of subspaces V1,…Vk <= V

Answer 1

Sum of subspaces V1,…,Vk <= V is set
( v1+…+ vk vi in Vi)

Question 2

What is sum of subspaces V1+…+Vk and list V1,…,Vk

Answer 2

Sum of subspaces V1+…+Vk is <= V (smallest subspace of V containing each Vi)
List V1,…,Vk is <= V1+…+Vk

Question 3

What is a direct sum

Answer 3

Direct sum is:
If V1,…,Vk <= V. Sum V1+…+Vk is direct if each v in V1 +…+Vk can be written as v = v1 +…+vk for unique vi in V.
V = u + w for unique u and w for all v

Question 4

When is a sum direct

Answer 4

A sum is direct when V1 N V2 = 0

Question 5

What is an internal direct sum

Answer 5

An internal direct sum is
V1 + V2 = V
V = V1 + V2, V1 N V2 = 0

Question 6

Why can 2 planes never be a direct sum

Answer 6

2 planes can never be a direct sum because if they intersect, they will do so in a line, so not just 0 in intersection

Question 7

When is a sum of vector subspace a direct sum

Answer 7

Sum of vector subspace is a direct sum when :
Vi N sumVj = 0 for V1,…,Vk

Question 8

When is a linear map a projection and example

Answer 8

A linear map pi is a projection when pi o pi = pi (applying twice is the same as applying it once) E.g mapping to X-axis

Question 9

If pi is a projection then what is V

Answer 9

V is ker(pi) +o im(pi)

Question 10

If V = W +o Uthen what are kernel and image of a projection pi : V to V

Answer 10

If V = W +o U then ker(pi): V to V is:
Ker(pi) = W
Im(pi) = U