SOLUTION: Find a basis for the subspace of R^3 consisting of all vectors [x1 x2 x3] such that −2x1−7x2−4x3=0.

Question 1159894: Find a basis for the subspace of R^3 consisting of all vectors [x1 x2 x3]
such that −2x1−7x2−4x3=0.
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.


Let  "p"  and  "q"  be two "free" parameters, i.e. arbitrary real numbers.


Consider the vectors  u = (-7p,2p,0)  and  v = (-2q,0,q).


They both belong to the given subspace in .


    Indeed, for  "u" :   -2*(-7p) - 7*(2p) - 4*0 = 14p - 14p - 0 = 0;

    and     for  "v" :   -2*(-2q) - 7*0 - 4*q    =  4q -  0 - 4q = 0.



Also, it is OBVIOUS that these vectors are LINEARLY INDEPENDENT.


Therefore, they form the basis in the given 2D subspace.

Solved.

"u" and "v" are the desired vectors.

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