3.1.2 Vector Space Axiom Definiti Let v be a set on which the operations of addition and scalar multiplication are (1).Ifx,y∈, then x+y∈, (2).Ifx∈ V and a is a scalar, then ax∈v The set v together with the operations of addition and scalar multiplication said to form a vector space if the following axioms are satisfied (a) x+y=y+x fo (b).(x+y)+z=x+(y+z) (c). There exist an element 0 in V such that x+0=x for each E V (d). For each x∈ 0 (e). a(x+y)=ax+ay for each real number a and any x and y in v (f).(a+ B)x=ax+ Bx for any real number a and B and any xE V (g).(aB)x=a(Bx)for any real number a and B and any xE V (h).1 The two-dimensional plane is the set of all vectors with two real-valued coordi nates. We label this set R2. It has two important properties (1).R is closed under scalar multiplication; every scalar multiple of a vector in the plane is also in the plane (2). R is closed under addition; the sum of any two vectors is always a vector in the plane 3.2 Linear Combination of vectors and basis vectors Definition A set of vectors in a vector is a basis for that vector space if any vector in the vector space can be written as a linear combination of them Example Any pair of two dimensional vectors that point in different directions will form3.1.2 Vector Space Axiom Definition: Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that (1). If x, y ∈ V, then x + y ∈ V, (2). If x ∈ V and α is a scalar, then αx ∈ V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied: (a). x + y = y + x for any x and y in V. (b). (x + y) + z = x + (y + z). (c). There exist an element 0 in V such that x + 0 = x for each x ∈ V. (d). For each x ∈ V, there exist an element −x ∈ V such that x + (−x) = 0. (e). α(x + y) = αx + αy for each real number α and any x and y in V. (f). (α + β)x = αx + βx for any real number α and β and any x ∈ V. (g). (αβ)x = α(βx) for any real number α and β and any x ∈ V. (h). 1 · x = x for all x ∈ V. Example: The two-dimensional plane is the set of all vectors with two real-valued coordi￾nates. We label this set R 2 . It has two important properties. (1). R 2 is closed under scalar multiplication; every scalar multiple of a vector in the plane is also in the plane. (2). R 2 is closed under addition; the sum of any two vectors is always a vector in the plane. 3.2 Linear Combination of Vectors and Basis Vectors Definition: A set of vectors in a vector is a basis for that vector space if any vector in the vector space can be written as a linear combination of them. Example: Any pair of two dimensional vectors that point in different directions will form 8