The document discusses the concepts of linear dependence and independence in vector spaces, defining a vector space as a set closed under vector addition and scalar multiplication. It illustrates linear independence with examples and describes how to determine whether a set of vectors is linearly dependent or independent using matrix methods. Key terms and processes for identifying linear dependence relations among vectors are outlined.

1. LINEAR DEPENDENCE &
INDEPENDENCE VECTORS

2. Presented By
1. Ashraful Islam Talukdar- 161-15-7100
2. Md Rakib Hossain- 161-15-6802
3. Fowjael Ahamed – 161-15-7045

3. VECTOR SPACES
 Definition: A vector space V is a set that is closed under finite vector
addition and scalar multiplication.
• An operation called vector addition that takes two vectors v, w ∈ V , and
produces a third vector, written v + w ∈ V .
• An operation called scalar multiplication that takes a scalar c ∈ F and a
vector v ∈ V , and produces a new vector, written cv ∈ V.

4. LINEAR INDEPENDENCE VECTORS
 Definition: An indexed set of vectors {v1, …, vp} in
is said to be linearly independent if the vector equation
has only the trivial solution. The set {v1, …, vp} is said to be linearly
dependent if there exist weights c1, …, cp, not all zero, such that,
----(1)
1 1 2 2
v v ... v 0p p
x x x   
1 1 2 2
v v ... v 0p p
c c c   

5. LINEAR INDEPENDENCE VECTORS
 Equation (1) is called a linear dependence relation
among v1, …, vp when the weights are not all zero.
 An indexed set is linearly dependent if and only if it is
not linearly independent.
 Example 1: Let , , and .1
1
v 2
3
 
 
 
  
2
4
v 5
6
 
 
 
  
3
2
v 1
0
 
 
 
  

6. How to Calculate it
Step 1
Construct a
matrix by
using Vectors
Step 2
Find REF.
Step 3
If REF has
no zero row.
LI

7. LINEAR DEPENDENCE VECTORS
 Definition: A finite set S = {x1, x2, . . . , xm} of vectors
in Rn is said to be linearly dependent if there exist
scalars (real numbers) c1, c2, . . . , cm, not all of which
are 0, such that
c1x1 + c2x2 + . . . + cmxm = 0.

8. LINEAR DEPENDENCE VECTORS
 Any set containing the vector 0 is linearly dependent, because for any c
6= 0, c0 = 0. 3.
 In the definition, we require that not all of the scalars c1, . . . , cn are 0.
The reason for this is that otherwise, any set of vectors would be
linearly dependent.
 Example 1: Let , 𝑣2 =
1
−1
2
and 𝑣3 =
3
1
4
.𝑣1 =
1
1
1

9. How to Calculate it
Step 1
Construct a
matrix by
using Vectors
Step 2
Find REF.
Step 3
If REF has
zero row.
LD

10. LINEAR DEPENDENCE RELATION
To check a set is LD
or LI
If set is LD
Then LDR is possible

11. MATHEMATICAL PROBLEMS

12. MATHEMATICAL PROBLEMS