"For what value of λ the following vectors..." was answered by our community

College
Algebra
Linear algebra
Vectors and spaces

Aaden Lam

Answered question

2022-01-21

For what value of

λ

the following vectors will form a basis for

E^{3}

a_{1} = (1, 5, 3), a_{2} = (4, 0, λ), a_{3} = (1, 0, 0)

Answer & Explanation

Brenton Pennington

Beginner2022-01-22Added 15 answers

$λ \in R - {0}$ .
Explanation:
Let the set $B = {a_{1} = (1, 5, 3), a_{2} = (4, 0, λ), a_{3} (1, 0, 0)}$
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.
$\exists$ unique $l, m, n \in R$ , s.t., $v = l a_{1} + m a_{2} + n a_{3}$ .
Now, $v = l a_{1} + m a_{2} + n a_{3}; l, m, n \in R$
$\Rightarrow (a, b, c) = l (1, 5, 3) + m (4, 0, λ) + n (1, 0, 0)$ , i.e.,
$(a, b, c) = (l, 5 l, 3 l) + (4 m, 0, m λ) + (n, 0, 0)$ , or,
$(a, b, c) = (l + 4 m + n, 5 l = b, 3 l = m λ = c$ .
In order that this system of eqns. may have a unique soln.,we know from Algebra that,
$∴ 1 (0) - 4 (0) + 1 (5 λ - 0) \neq 0$
$∴ λ \neq 0$
Hence, $λ$ can be any non zero real number.

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