Question

Show that {(1,1,0),(1,0,1),(0,1,1)} is linearly independent subset of r^3

Answer 1

Answer:  Yes, the given set of vectors is a linearly independent subset of R³.

Step-by-step explanation:  We are given to show that the following set of three vectors is a linearly independent subset of R³ :

B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} .

Since the given set contains three vectors which is equal to the dimension of R³, so it is a subset of R³.

To check the linear independence, we will find the determinant formed by theses three vectors as rows.

If the value of the determinant is non zero, then the set of vectors is linearly independent. Otherwise, it is dependent.

The value of the determinant can be found as follows :

$D\\\\\\=\begin{vmatrix}1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1\end{vmatrix}\\\\\\=1(0\times1-1\times1)+1(1\times0-1\times1)+0(1\times1-0\times0)\\\\=1\times(-1)+1\times(1)+0\\\\=-1-1\\\\=-2\neq0.$

Since the determinant is not equal to 0, so the given set of vectors is a linearly independent subset of R³.

Thus, the given set is a linearly independent subset of R³.