LINEAR COMBINATION OF VECTORS

LINEAR COMBINATION OF VECTORS

Let V(F) be a vector space. A vector x ∈ V is said to be linear combination of vectors x₁, x₂, x₃,.....xₙ ∈ V if x = a₁x₁ + a₂x₂ + .....+ aₙxₙ

where a₁, a₂, .......,aₙ ∈ F.

A linear combination of vectors is a sum of vectors where each vector is multiplied by a scalar coefficient (also known as a scalar multiple). The scalar coefficients are also known as weights or multipliers and they determine how much of each vector is included in the linear combination. The resulting vector is also a vector in the same space as the original vectors.

The mathematical notation for a linear combination of vectors is:

v = c1v1 + c2v2 + ... + cn*vn

where v is the resulting vector, c1, c2, ..., cn are the scalar coefficients, and v1, v2, ..., vn are the original vectors.

More about this.

A linear combination can also be represented using matrix notation. The vector v can be represented as a column vector, and the vectors v1, v2, ..., vn can be represented as column vectors in a matrix called the "coefficient matrix". The scalar coefficients c1, c2, ..., cn can be represented as elements in a column vector called the "weight vector".

The linear combination can then be represented as the matrix product of the coefficient matrix and the weight vector:

v = A * w

where A is the coefficient matrix and w is the weight vector.

Linear combinations of vectors have many applications in various fields such as physics, engineering, computer science and mathematics. For example, in physics, a linear combination of vectors can be used to represent the position of a particle in space, and in computer graphics, linear combinations of vectors are used to represent transformations of objects in 3D space.

Linear combinations of vectors are also used in the field of linear algebra, for example, to find the solution to a system of linear equations, or to find the eigenvectors of a matrix.

Example 1:

Express ( 1, 2, 3 ) as a linear combination of (1, 1, 1), (2, -1, 1) and (1, -2, 5) in R³(R).

Solution:

( 1, 2, 3 ) = a₁(1, 1, 1) + a₂(2, -1, 1) + a₃(1, -2, 5) 

for a₁, a₂, a₃ ∈ R

= ( a₁ + 2a₂ + a₃, a₁ - a₂ -2a₃, a₁ - a₂ +5a₃)

then a₁ + 2a₂ + a₃ = 1, a₁ - a₂ -2a₃ = 2 ,  

a₁ - a₂ +5a₃ = 3

Solving these equations we get 

a₁ = 2, a₂ = -2/3 , a₃ = 1/3

Example 2:

The vector (2, -5, 4) cannot be expressed as a linear combination of ((1, -3, 2) and (2, -1, 1) in R³(R).

SMALLEST SUBSPACE CONTAINING A SUBSET 

Let S be subset of a vector space V. Then, a subset U of V is called the smallest subspace containing S, if U is a subspace of V containing S and is itself contained in every subspace of V containing S. Therefore, the intersection of all subspaces of V containing S is also a subspace of V. This subspace is called the subspace of V generated or spanned by S and is denoted by [S].

If {Wᵢ}ᵢ∈I be a family of all subspaces of V containing S, then

 [S] = ∩ᵢ∈I W ⊆ Wᵢ ∀ i ∈ I.

LINEAR SPAN 

Let V(F) be a vector space and S be any non-empty subset of V. Then the linear span of set S is the set of all linear combination of finite subsets of elements if S and is denoted by L[S] or [S]. Thus, 

L(S) = {x:x = ∑ αᵢxᵢ , αᵢ ∈ F, xᵢ ∈ S } 

L(S) is also called the set spanned or generated by S. 

Theorem 1: 

The linear span L(S) of any subset S of a vector space V(F) is a subspace of V generated by S.

Proof. Let x,y be any two elements of L(S)

Then, x = α₁x₁ , + α₂x₂ + ....+ αₙxₙ = ∑ αᵢxᵢ  

y = β₁y₁ + β₂y₂ + β₃y₃ ....+ βy =∑ βᵢyᵢ

where αᵢ βᵢ ∈ F and xᵢ yᵢ ∈ S 

Now if a, b ∈ F, then 

ax + by = ∑aαᵢxᵢ + ∑bβᵢyᵢ ∈ L(S)

Since ax + by is a linear combination of finite sets 

x₁, x₂, ....,xₙ ; x₁, x₂, .......yₙ of elements of S.

Thus a, b ∈ F and x,y ∈ L(S) ⟹ ax + by ∈ L(S)

Hence L(S) is a subspace of V(F) 

Also each element of S belongs to L(S),

because if xᵢ ∈ S, then xᵢ = 1xᵢ and this implies xᵢ ∈ L(S). Thus L(S) is a subspace of V(F) and S is contained in L(S). 

Now if U is any subspace of V containing S, then each element of L(S) must be in U because U is to be closed under vector addition and scalar multiplication. Therefore L (S) will be contained in U.