determine whether the given set of vectors generate the vect

determine whether the given set of vectors generate the vector space dice

Determine whether the given set of vectors generate the vector space dice En R^3: (1, -1, 2), (-1, 1, 2), (0, 0, 1)

Solution

A set of n vectors span a n-dimensional vector space if this set is a basis, and this is done if the nnvectors are linearly independent.

that is

a1v1+a2v2 +...........+anvn=0

In your case you have three vectors

then it is a1v1+a2v2+a3v3=0

a1(1,-1,2) + a2(-1,1,2) +a3(0,0,1)=0

so now we get three equations

a1-a2+0a3 =0

-a1 +a2+0a3=0

2a1+2a2+a3=0

so a1=a2

a3= -4a1

so we get solution has (0,0,0)

So we have, in fact, shown linear independence. And any set of three linearly independent vectors in R^3 spans R^3. Hence our set of vectors is indeed a basis for R^3.