Matematika Diskrit : Relasi, Sifat Relasi, dan Contoh Soal

Matriks dan Ruang Vektor : Ruang Vektor Umum dan Subspace Vektor

Real Vector Spaces

Definition

V is a vector space if the following axioms are satisfied by all objects u,v,w in V and all scalar k and m.

1. If u,v∈V, then u+v∈V
2. u+v=v+u
3. u+(v+w)=(u+v)+w
4. There is an object 0 in V such that 0+u=u+0=u
5. For each u∈V, there is -u∈V such that u+(-u)=(-u)+u=0
6. If k is any scalar and u is any object in V, then ku∈V
7. k(u+v)=ku+kv
8. (k+m)u=ku+mu
9. k(mu)=(km)u
10. 1u=1

Example 1

• R^n
• M_(m×n)
• etc

Subspaces Vectors

Definition

If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisfied

• If u and v are vectors in W, then u+v ∈W
• If k is a scalar and u is a vector in W, then ku ∈W

A subspace is a vector space

Example 2

1. All vectors of the form (a,0, 0) is a subspace of R^3
2. All vectors of the form (a,b,c) where b = a+c is a subspace of R^3

Is all vectors of the form (a, b, c) where b = a+c+1 a subspace of R^3?

Linear Combination

w is a vector in a vector space V. w is said to be a linear combination on the vectors v_1,v_2,…,v_r in V if w can be expressed in the form

w = k_1 v_1+k_2 v_2+…+k_r v_r

Where k_1,k_2,…,k_r are scalars. These scalars are called the coefficients of the linear combination

Ex :

Consider the vectors u = (1, 2, -1) and v = (6, 4, 2) in R^3

Show that :

1.w = (9, 2, 7) is a linear combination of u and v
2.w′ = (4, -1, 8) is not a linear combination of u and v

Span

S={w_1,w_2,…,w_r} is a nonempty set of vectors in a vector space V. The subspace W of V that consists of all possible linear combinations of the vectors in S is called the subspace of V generated by S. The vectors w_1,w_2,…,w_r are called span W

Denote :

Ex :

1. e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1) span R^3?
2. v_1= (1,1,2), v_2 = (1,0,1), v_3 = (2,1,3) span R^3?

Linear Independence

Definition

S = {v_1,v_2,…,v_r} is a set of two or more vectors in a vector space V. S is said to be a linearly independent set if no vector in S can be expressed as a linear combination of the others or

k_1 v_1 + k_2 v_2 +…+ k_r v_r = 0 

Are k_1 = 0, k_2 = 0,…, k_r = 0 

A set that is not linearly independent is said to be linearly dependent

Example 3

Determine whether the vectors

v_1=(1, -2, 3), v_2=(5, 6, -1), v_3=(3, 2, 1) 

Are linearly independent or linearly dependent in R^3! 

Determine whether the vectors

v_1 = (1, 2, 2, -1), v_2 = (4, 9, 9, -4), v_3 = (5, 8, 9 ,-5) 

In R^4 are linearly dependent or linearly independent!