Direct Sums 2

Question 1

If V = V1 +o V2, then what is dim V

Answer 1

If V = V@+o V2, dim V = dimV1 + dim V2 (if finite dimensional)

Question 2

What is dim V1 +o … +o Vk for V1+…+Vk direct

Answer 2

Dim V1 +o … +o Vk = sum i = 1 to k dimVi

Question 3

If V1,V2 have bases B1: v1…vk and B2: w1,…wk , when is sum V1+V2 direct

Answer 3

If V1,V2 have bases B1: v1,…Vk and B2: w1,…wk then sum V1 + V2 is direct when:
Concatenation B1,B2: v1,…,vk,w1,…,wk is a basis of V1 + V2 (this is spanning), so check whether L.I

Question 4

When is sum =V1+…+Vk direct with V1,…,Vk with bases B1,…,Bk

Answer 4

Sum V1+…+Vk is direct with V1,…,Vk with bases B1,…,Bk when B1,B2,…,Bk is a basis of V1+…+Vk

Question 5

What is the complement

Answer 5

Complement is:
V1 +o V2 +o … +o Vk = V (internal direct sum) then V2,V3…,Vkis complement of V1 and vice versa)