Question

Determine all values of the constant k for which the given set of vectors is linearly independent

Answer 1

All values of the constant k for which the given set of vectors is linearly independent if,

Linearly Independent Vectors:

Given four vectors in the four-dimensional Euclidean space, the vectors are linearly independent if and only if the determinant of the four by four matrix with these vectors as its rows is not zero.

The given vectors are linearly independent in R4 if and only if

⇒

$det \left[\begin{array}{cccc}1&1&0&-1\\1&k&1&1\\5&1&k&1\\-1&1&1&k\end{array}\right] \neq 0$

⇒ we solve this inequality as follows:

$det \left[\begin{array}{cccc}1&1&0&-1\\1&k&1&1\\5&1&k&1\\-1&1&1&k\end{array}\right] = (1) det \left[\begin{array}{ccc}k&1&1\\1&k&1\\1&1&k\end{array}\right] - (1) det\left[\begin{array}{ccc}1&1&1\\5&k&1\\-1&1&k\end{array}\right] + (0) det\left[\begin{array}{ccc}1&k&1\\5&1&1\\-1&1&k\end{array}\right] - (-1) det \left[\begin{array}{ccc}1&k&1\\5&1&k\\-1&1&1\end{array}\right]$  

⇒ $(k(k^2 - 1) - 1 (k-1) + (1-k) ) - ((k^2 -1 ) - (5k+1) + (5+k)) + ((1-k) - k(5+k) + (5+1))$

⇒ k³ - 2k³ -5k + 6 = (k - 3)(k - 1)(k + 2) $\neq$ 0 ,

This implies that for any constant k ∈ R - {-2, 1, 3}, the four given vectors are linearly independent.

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