The values of x for which the given vectors are basis for R³ is:
[tex]x\neq 1[/tex]
We know that for a set of vectors are linearly independent if the matrix formed by these set of vectors is non-singular i.e. the determinant of the matrix formed by these vectors is non-zero.
We are given three vectors as:
(-1,0,-1), (2,1,2), (1,1, x)
The matrix formed by these vectors is:
[tex]\left[\begin{array}{ccc}-1&2&1\\0&1&1\\-1&2&x\end{array}\right][/tex]
Now, the determinant of this matrix is:
[tex]\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-1(x-2)-2(1)+1\\\\\\\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-x+2-2+1\\\\\\\begin{vmatrix}-1 &2 & 1\\ 0& 1 & 1\\ -1 & 2 & x\end{vmatrix}=-x+1[/tex]
Hence,
[tex]-x+1\neq 0\\\\\\i.e.\\\\\\x\neq 1[/tex]
