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STatiana [176]
3 years ago
7

For waht values of x do the vectors -1,0,-1), (2,1,2), (1,1, x) form a basis for R3?

Mathematics
1 answer:
DaniilM [7]3 years ago
4 0
<h2>Answer:</h2>

The values of x for which the given vectors are basis for R³ is:

                        x\neq 1

<h2>Step-by-step explanation:</h2>

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:

\left[\begin{array}{ccc}-1&2&1\\0&1&1\\-1&2&x\end{array}\right]

Now, the determinant of this matrix is:

\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

Hence,

-x+1\neq 0\\\\\\i.e.\\\\\\x\neq 1

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