Inter section of subspaces Union of subspaces Linear Sum of subspaces Linear Span of a set Linear Dependence of vector & Linearly Dependent set (LD set) Linear Independence of vector & Linearly Independent set (LI set)

2. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
Inter section of subspaces
Union of subspaces
Linear Sum of subspaces
Linear Span of a set
Linear Dependence of vector & Linearly Dependent set (LD set)
Linear Independence of vector & Linearly Independent set (LI set)

3. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐

4. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐 𝟑 𝐧
𝟏 𝟐 𝟑 𝐧 𝒊
𝒏
𝒊 𝟏

5. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐
SEE
SEE

6. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

7. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

8. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐
𝟏 𝟏 𝟐 𝟐 𝟏 𝟐

9. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐
𝟏 𝟏 𝟐 𝟐 𝟏 𝟐

10. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐 𝟏 𝟐
𝟏 𝟏 𝟐 𝟐
𝟏 𝟐

11. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

12. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

𝒊 𝒊 𝒊 𝒊 .





13. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

14. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
.
L(S)
S

15. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
Example 2:-
Solution:-
{ which is required Linear combination }

16. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

17. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
Example 3:
3
.

18. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐
𝟏 𝟐 𝟏 𝟐.

19. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
)
)
)
)
𝟏 𝟐 𝟏 𝟐.

20. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

21. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

22. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

23. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

24. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

25. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe

26. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐 𝟑 𝒏

27. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐 𝟑 𝐧
𝟏 𝟐 𝟑 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐧 𝐧
𝟏 𝟐 𝟑 𝐧 𝟏 𝟐 𝟑 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐧 𝐧 𝟏 𝟐 𝟑 𝐦

28. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐 𝟑 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤
𝟏 𝟐 𝟑 𝐤
𝟏 𝟐 𝟑 𝐤
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤 𝐤 𝟏 𝐧

29. MANIKANTA SATYALA || || VECTOR SPACES
AlgebrA Sub-SPACe
𝟏 𝟐 𝟑 𝐧

30. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES

31. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES

32. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES

33. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES

34. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES
𝟏 𝟐 𝟑 𝐧
𝐤
𝟏 𝟐 𝟑 𝐤 𝐧
𝟏 𝟐 𝟑 𝐤 𝐧
𝟏 𝟐 𝟑 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤 𝐧 𝐧
𝟏 𝟐 𝟑 𝐤 𝐧
𝐤
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤 𝐤 𝟏 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤
𝟏 𝟏
𝑩𝒖𝒕 𝒂 𝟏 ≠ 𝟎, 𝒔𝒐 𝜶 𝟏 = 𝟎 , 𝒘𝒉𝒊𝒄𝒉 𝒊𝒔 𝒄𝒐𝒏𝒅𝒓𝒂𝒅𝒊𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒆𝒂𝒄𝒉 𝒆𝒍𝒆𝒎𝒆𝒏𝒕 𝒐𝒇 𝑺 𝒊𝒔 𝒂 𝒏𝒐𝒏𝒛𝒆𝒓𝒐 𝒗𝒆𝒄𝒕𝒐𝒓

35. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝐤
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝟏 𝐤 𝟏 𝐤 𝐤
𝐤 𝐤 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝟏 𝐤 𝟏
𝐤 𝐤
𝟏
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐤 𝟏 𝐤 𝟏
𝐤 𝐤
𝟏
𝟏 𝟏 𝐤
𝟏
𝟐 𝟐 𝐤
𝟏
𝟑 𝟑 𝐤
𝟏
𝐤 𝟏 𝐤 𝟏
𝐤
𝒑
𝟏 𝟐 𝟑 𝒑 𝟏
𝒑 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐩 𝟏 𝐩 𝟏

36. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES
𝒑 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐩 𝟏 𝐩 𝟏
𝒑 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐩 𝟏 𝐩 𝟏
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐩 𝟏 𝐩 𝟏 𝒑
𝟏 𝟐 𝟑 𝒑
𝟏 𝟐 𝟑 𝒑 𝐧

37. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES
𝟏 𝟐 𝟑 𝐧
𝐢
𝟏 𝟐 𝟑 𝐢 𝟏 𝐢 𝟏 𝐧
𝟏 𝟐 𝟑 𝐧
𝟏 𝟐 𝟑 𝐢 𝟏 𝐢 𝟏 𝐧
𝟏 𝟐 𝟑 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐢 𝐢 𝐧 𝐧
𝐢 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐢 𝟏 𝐢 𝟏 𝐢

38. AlgebrA Sub-SPACe
MANIKANTA SATYALA || || VECTOR SPACES
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐢 𝐢 𝐧 𝐧
𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐢 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 𝐢 𝟏 𝐢 𝟏 𝐧 𝐧
𝟏 𝐢 𝟏 𝟏 𝟐 𝐢 𝟐 𝟐 𝟑 𝐢 𝟑 𝟑
𝐢 𝟏 𝐢 𝐢 𝟏 𝐢 𝟏 𝐢 𝟏 𝐢 𝟏 𝐧 𝐧