For what value of \lambda the following vectors will form

Aaden Lam 2022-01-21 Answered
For what value of λ the following vectors will form a basis for E3.
a1=(1,5,3),a2=(4,0,λ),a3=(1,0,0)?
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Expert Answer

Brenton Pennington
Answered 2022-01-22 Author has 15 answers

λR{0}.
Explanation:
Let the set B={a1=(1,5,3),a2=(4,0,λ),a3(1,0,0)}
form a Basis for the vector space E3.
Then an arbitrary vector v=(a,b,c)E3 can uniquely be represented as a linear combination of the vectors in B.
unique l,m,nR, s.t., v=la1+ma2+na3.
Now, v=la1+ma2+na3;l,m,nR
(a,b,c)=l(1,5,3)+m(4,0,λ)+n(1,0,0), i.e.,
(a,b,c)=(l,5l,3l)+(4m,0,mλ)+(n,0,0), or,
(a,b,c)=(l+4m+n,5l=b,3l=mλ=c.
In order that this system of eqns. may have a unique soln.,we know from Algebra that,
1(0)4(0)+1(5λ0)0
λ0
Hence, λ can be any non zero real number.

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