Problem-25

linear independence

Question

Choose the correct options.

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

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

  3. The set \(\{(1, 2, 3), (0, 5, 6), (0, 0, 9)\}\) is linearly independent.

  4. The set \(\{(1, 2, 3), (0, 0, 6), (0, 0, 9)\}\) is linearly independent.

Solution

Options (a) and (c) are correct. Here is an algorithm that will work for any set of vectors:

  • Add the vectors as the rows of a matrix.
  • If the rank of the resulting matrix is equal to the number of rows, then the vectors are linearly independent.
  • If the rank of the matrix is less than the number of rows, then the vectors are linearly dependent.

Option-(a) \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & -3 & -6\\ 7 & 8 & 9 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & -3 & -6\\ 0 & -6 & -12 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 2\\ 0 & 1 & 2 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 2\\ 0 & 0 & 0 \end{bmatrix} \] The vectors are linearly dependent.

Option-(b) \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 5 & 7 & 9 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & -3 & -6\\ 5 & 7 & 9 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & -3 & -6\\ 0 & -3 & -6 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 2\\ 0 & 0 & 0 \end{bmatrix} \] The vectors are linearly dependent.

Option-(c) \[ \begin{bmatrix} 1 & 2 & 3\\ 0 & 5 & 6\\ 0 & 0 & 9 \end{bmatrix} \] This is clearly linearly independent.

Option-(d)

\(\{(1, 2, 3), (0, 0, 6), (0, 0, 9)\}\) is dependent. We can see that the third and second vectors are just multiples of each other.