Question-23

Choose the correct options.

• The set \(\{(1, 2, 3), (4, 5, 6), (7, 8, 9)\}\) is linearly dependent.
• The set \(\{(1, 2, 3), (4, 5, 6), (5, 7, 9)\}\) is linearly independent.
• The set \(\{(1, 2, 3), (0, 5, 6), (0, 0, 9)\}\) is linearly independent.
• The set \(\{(1, 2, 3), (0, 0, 6), (0, 0, 9)\}\) is linearly independent.

Answer

• The set \(\{(1, 2, 3), (4, 5, 6), (7, 8, 9)\}\) is linearly dependent.
• The set \(\{(1, 2, 3), (4, 5, 6), (5, 7, 9)\}\) is linearly independent.
• The set \(\{(1, 2, 3), (0, 5, 6), (0, 0, 9)\}\) is linearly independent.
• The set \(\{(1, 2, 3), (0, 0, 6), (0, 0, 9)\}\) is linearly independent.

Solution

Option-1

We have three vectors in \(\mathbb{R}^{3}\). Let us form the matrix with these three vectors as rows: \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \] We can now perform row reduction: \[ \begin{equation*} \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\xrightarrow{R_{2}\rightarrow R_{2} -R_{1}}\begin{bmatrix} 1 & 2 & 3\\ 3 & 3 & 3\\ 7 & 8 & 9 \end{bmatrix}\xrightarrow{R_{3}\rightarrow R_{3} -R_{1}}\begin{bmatrix} 1 & 2 & 3\\ 3 & 3 & 3\\ 6 & 6 & 6 \end{bmatrix} \end{equation*} \] Note that the third row is a multiple of the second row. Hence the determinant is zero. The vectors are linearly dependent.

Option-2

The third vector in the set is the sum of the first two. Hence this set is linearly dependent.

Option-3

Adding these three vectors as the rows of matrix, we see that the matrix has full rank. Hence, the set is linearly independent.

Option-4

The third vector is a multiple of the second. Hence this set is linearly dependent.