Linear dependence(ld) &linear independence(li)

complex variable & numerical method

1. VECTOR CALCULUS AND LINEAR ALGEBRA (2110015) Made By :- DIVYANGSINH RAJ (150990119004) NAVED FRUITWALA (150990119006) UTKARSH GANDHI (150990119007) Branch :- MECHANICAL ENGINEERING Semester :- 2 ᴺᴰ

2. TOPIC :- LINEAR DEPENDENCE(LD) & LINEAR INDEPENDENCE(LI )

3. Linear Dependence and Independence :- A finite set of vector of a vector space is said to be Linearly Dependent(LD) if there exists a set of scalars k 1 , k 2 , ……, k n such that, k 1 u 1 + k 2 u 2 +……+ k n u n = ō A finite set of vector of a vector space is said to be Linearly Independent(LI) if there exists scalars k 1 , k 2 , ……, k n such that, k 1 u 1 + k 2 u 2 +……+ k n u n = ō => k 1 = k 2 = ……= k n =0

4. Properties For LI - LD Property 1: Any subset of a vector space is either L.D. or L.I. Property 2: A set of containing only ō vector that is {ō} is L.D. Property 3: A set is containing the single non zero vector is L.I. Property 4: A set having one of the vector as zero vector is L.D.

5. EXAMPLES 1.] Consider the set of vectors to check LI or LD {(1,0,0),(0,1,0),(0,0,1)} in R ³. Solution :- Let k ₁,k₂,k ₃ belongs to R such that, k ₁(1,0,0)+k₂(0,1,0)+k₃(0,0,1)=(0,0,0) ( k ₁,k₂,k ₃ )=(0,0,0) k ₁=0,k₂=0,k₃=0 Therefore, the set { i,j,k } is LI.

6. 2.] Determine whether the vectors are LI in R ³ (1-2,1),(2,1,-1),(7,-4,1). Solution :- Let k ₁,k₂,k ₃ belongs to R such that, k ₁(1,-2,1)+k₂(2,1,-1)+k₃(7,-4,1)=(0,0,0) k ₁+2k₂+7k₃=0 -2k ₁+1k₂-4k₃=0 k ₁-k₂+k₃=0

7. |A|= |A|= 1[(1)(1)-(-4)(-1)] -2[(-2)(1)-(-4)(1)] +7[(-2)(-1)-(1)(1)] |A|= -3-4+7 | A|=0 Since the determinant of the system is zero, the system of these equations has a nontrivial solution. That is at least one of k ₁,k₂,k ₃ is nonzero. Thus the vectors are LD.  

9. Solution:- i .) 2-x+ , 3+6x+2 , 2+10x- Let, + + =0 => (2-x+ )+ + = 2 +3 +2 =0 - +6 +10 =0 4 +2 =0  

10. ~ = ~ A = = 2(-24-20)-3(4-40)+2(-2-24) = -88+108-52      

11. = -32 ǂ 0 Therefore the system has unique solution The given vectors are not L.D(i.e they are L.I).  

12. Solution:- ii.) A= 2+x+ , x+ , 2+2x+ Let, + + = + )+ )=0 2 +0 +2 =0 + + 2 =0 +2 =0  

13. ~ A= ~ = = 2(3-4) + 2(2-1) =2(-1) + 2(1) =0      

14. = 0 Therefore the system has infinitely many solution The given vectors are L.D(i.e they are not L.I).  

15. THANK YOU