Reviewing notes 1.1 of Analytic geometry

Chapter 1 Vector Algebra

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♦ Vector Space

Vector and vector space

A vector is described as a quantity that has both direction and length.  A vector space is a collection of these geometic objects that can be added together and multiplied by numbers.

What we mainly focus on is the vector space over the real number field,which is a nonempty set V together with two operations called addition and scalar multiplication.The sum u+v of two elements u,v∈V is also an element of V,and the scalar multiple cu of u∈V by the real number c is an element of V. These operations are required to satisfy the axioms followed.

Axiom

Let V be a vector space over the real number field:

(a)Addition is associative and commutative.

(b)There is a zero element θ such that u+θ=u for every u∈V.

(c)The distribution laws hold:

(c+d)u=cu+du,c(u+v)=cu+cv;

for every real numbers c,d and u,v∈V.

(d)(cd)u=c(du) for every real c,d,and u∈V.

(e)0u=θ,1u=u,for every u∈V.

Example

(a) In a vector space the additive inverse −u is often called the opposite vector of u; it has the same magnitude as the original and opposite direction and we have u+(-u)=θ and -(-u)=u.A unit vector in a normed vector space is a vector of length 1.The normalized vector û of a non-zero vector u is the unit vector in the direction of u.

(b) In Euclidean space, two vectors are orthogonal if and only if their scalar product is zero, or one of the vectors is zero. It is an extension of the concept of perpendicular vectors to spaces of any dimension.

Definition

A subset B of a vector space V is called a linearly dependent set if there exist distinct elements u₁,u₂...,u_m∈B And real numbers c¹,c²...c^m not all 0,such that c¹u₁ +c²u₂ +...+c^mu_m=θ.If B is not linearly dependent,then it is an linearly independent set.V is a finite dimensional vector space if some finite subset B of V spans V,namely every element u∈V is a linear combination u=c¹u₁+c²u₂+...+c^mu_m where u₁,u₂...,u_m∈B. If u₁,u₂...,u_m are linearly independent,then the combination is unique ,and we call the linearly independent set {u1,u2...,um} that spans V a basis for V.

Proposition

Vectors α,β,γ are coplanar if and only if there exist three real numbers λ,μ,ν such that λα+μβ+νγ=θ.

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