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) # ?

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Answer 1 · 2017-12-20T16:42:59

# lambda in RR-{0}#.

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.