Group members : Chin Yong Keat, Lee Lai Kuan, Lim Pei Ying, Ling Yu Hui

Linear Dependence

An indexed set of vectors {v1 , v2 ,…v p } in V is said to be linearly independent if the

vector equation c1 v1 + c2 v2 + … + c p v p = 0 has only the trivial solution, c1 = 0,…, c p =

01.

The set {v1 , v2 ,…v p } is said to be linearly dependent if c1 v1 + c2 v2 + … + c p v p = 0has

a nontrivial solution.

Example:

Let p1(t) = 1, p2(t) = t, and p3(t) = 4 – t. Then { p1 , p2 , p3 } is linearly dependent in P

because p3 = 4p1 - p2.

We know that in the n-dimensional Euclidean space any vector can be written uniquely

as a linear combination of the n-coordinate unit vectors e1 = (1,0,…,0), e2 = (0,1…,0),

and en = (0,0,…,1). Thus, the n unit vectors determine all the vectors. This leads to the

definition as follow:

Let V be a vector space over F. A set {v1 , v2 ,…vn } of n elements is called a basis of V if

every v is elements from V can be expressed uniquely as v = a1 v1 + a1 vn + …. + an vn . It

is clear from the definition that if {v1 , v2 ,…vn } is a basis of V, the subspace generated by

them is the whole space V. Conversely if we are given a set of elements {v1 , v2 ,…vm } which generate the whole space V, they need not form a basis for V as the following

example shows.

Theorem: An indexed set {v1 , v2 ,…v p } of two or more vectors, with v1 ≠ 0, is linearly

dependent if and only if some v j (with j > 1) is a linear combination of the preceding

vectors, v1,…,v j−1.

Group members : Chin Yong Keat, Lee Lai Kuan, Lim Pei Ying, Ling Yu Hui

Example:

Let V = R3 and let v1 = (1,0,0), let v2 = (0,1,0), = let v3 = (0,0,1) and let v4 = (1,2,0). Then

v1,v2, v3 and v4 generate the space V, but they do not form a basis for V as the element

v = (1,1,1) can be expressed in terms of v1,v2, v3 and v4 in two different ways,

v = v1+v2 + v3 and v = −v2 + v3 + v4.