Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For thos

Braxton Pugh

Braxton Pugh

Answered question

2021-02-12

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. All vectors (x, y, z) in V3 whose components satisfy a system of three linear equations of the form:
a11x+a12y+a13z=0
a21x+a22y+a23z=0
a31x+a32y+a33z=0

Answer & Explanation

timbalemX

timbalemX

Skilled2021-02-13Added 108 answers

Let u=(x1,y1,z1) and v=(x2,y2,z2) be in VLE , the set of all vectors satisfying the given linear equations , and a be an arbitrarry real number
So that:
a11x1+a12y1+a13z1=0
a21x1+a22y1+a23z1=0
a31x1+a32y1+a33z1=0 and a11x2+a12y2+a13z2=0
a21x2+a22y2+a23z2=0
a31x2+a32y2+a33z2=0
Axiom 1: u+v=(x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2)VLE
whereas
a11(x1+x2)+a12(y1+y2)+a13(z1+z2)=0
a21(x1+x2)+a22(y1+y2)+a23(z1+z2)=0
a31(x1+x2)+a32(y1+y2)+a33(z1+z2)=0
Axiom 2: av=a(x1,y1,z1)=(ax1,ay1,az1)VLE while: a11(ax1)+a12(ay1)+a13(az1)=0
a21(ax1)+a22(ay1)+a23(az1)=0
a31(ax1)+a32(ay1)+a33(az1)=0
therefore VLE is real linear subspace since the last axioms are satisfies for the all vectors in V3

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