Question

Find all the values of k so that the set{ ( 1,−3,2),(−3,9,−6),(5,−7,k)} form the basis for R³

Answer 1

For a set to form a basis in R³ it is sufficient that the determinant $det \begin{vmatrix} 1 & -3 & 2 \\ -3 & 9 & -6 \\ 5 & -7 & k \end{vmatrix}$ was not equal to zero.

$det \begin{vmatrix} 1 & -3 & 2 \\ -3 & 9 & -6 \\ 5 & -7 & k \end{vmatrix} = 9\cdot k+ 2\cdot (-3)\cdot (-7) + (-3)\cdot (-6)\cdot 5 - 2\cdot 9 \cdot 5 - (-3) \cdot (-3) \cdot k - 1\cdot (-6)\cdot (-7) = 0$

So there is no such k that these three vectors form a basis.

It is clear from the fact, that the first two vectors are linear dependent as 3*(1, -3, 2) + 1*(-3,9, -6) = 0