Write a linear equation defining the subspace of R_3 spanned

horrupavinvo 2022-02-22 Answered
Write a linear equation defining the subspace of R3 spanned by v1=(0,0,2) and v2=(3,1,1).
(Write your answer in the form ax+by+cz. For example "2x+3y-4z")
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Expert Answer

lilwayne10j6o
Answered 2022-02-23 Author has 8 answers
Every element of your linear subspace can be written as a linear combination of v1 and v2. If (x1,y1,z1) and (x2,y2,z3) both satisfy
ax+by+cz=0
then so does any linear combination of the two. Since v1,v2 span a 2 dimensional subspace of a 3 dimensional space, we only need one equation to describe it, and both v1 and v2 must solve it. So we need
a0+b0+c2=0
and
a(3)+b(1)+c(1)=0
Solving these equations gives (a,b,c) up to some factor that we can choose, since
ax+by+cz=0
and
kax+kby+kcz=0
have the same solution set if k0
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