# For what value of \lambda the following vectors will form

For what value of $\lambda$ the following vectors will form a basis for ${E}^{3}$.
${a}_{1}=\left(1,5,3\right),{a}_{2}=\left(4,0,\lambda \right),{a}_{3}=\left(1,0,0\right)$?
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Brenton Pennington

$\lambda \in \mathbb{R}-\left\{0\right\}$.
Explanation:
Let the set $B=\left\{{a}_{1}=\left(1,5,3\right),{a}_{2}=\left(4,0,\lambda \right),{a}_{3}\left(1,0,0\right)\right\}$
form a Basis for the vector space ${E}^{3}$.
Then an arbitrary vector $v=\left(a,b,c\right)\in {E}^{3}$ can uniquely be represented as a linear combination of the vectors in B.
$\mathrm{\exists }$ unique $l,m,n\in \mathbb{R}$, s.t., $v=l{a}_{1}+m{a}_{2}+n{a}_{3}$.
Now, $v=l{a}_{1}+m{a}_{2}+n{a}_{3};l,m,n\in \mathbb{R}$
$⇒\left(a,b,c\right)=l\left(1,5,3\right)+m\left(4,0,\lambda \right)+n\left(1,0,0\right)$, i.e.,
$\left(a,b,c\right)=\left(l,5l,3l\right)+\left(4m,0,m\lambda \right)+\left(n,0,0\right)$, or,
$\left(a,b,c\right)=\left(l+4m+n,5l=b,3l=m\lambda =c$.
In order that this system of eqns. may have a unique soln.,we know from Algebra that,
$\therefore 1\left(0\right)-4\left(0\right)+1\left(5\lambda -0\right)\ne 0$
$\therefore \lambda \ne 0$
Hence, $\lambda$ can be any non zero real number.