For what value of lamda the following vectors will form a basis for E^3 a_1 = (1,5,3) , a_2 = (4,0,lamda) , a_3 = (1,0,0) ?

1 Answer
Dec 20, 2017

lambda in RR-{0}.

Explanation:

Let the set B={a_1=(1,5,3), a_2=(4,0,lambda),a_3=(1,0,0)}

form a Basis for the vector space E^3.

Then an arbitrary vector v=(a,b,c) in E^3 can uniquely be

represented as a linear combination of the vectors in B.

In other words,

EE" unique "l,m,n in RR," s.t., "v=la_1+ma_2+na_3.

Now, v=la_1+ma_2+na_3; l,m,n in RR,

rArr (a,b,c)=l(1,5,3)+m(4,0,lambda)+n(1,0,0), i.e., ,

(a,b,c)=(l,5l,3l)+(4m,0,mlambda)+(n,0,0), or,

(a,b,c)=(l+4m+n,5l,3l+mlambda).

By the equality of vectors, then, we have,

l+4m+n=a, 5l=b, 3l+mlambda=c.

In order that this system of eqns. may have a unique soln.,

we know from Algebra that,

|(1,4,1),(5,0,0),(3,lambda,0)| ne 0.

:. 1(0)-4(0)+1(5lambda-0) ne 0.

:. lambda ne 0.

Hence, lambda can be any non zero real number.