Problem
18. Three vectors $\underline{a}, \underline{b}$ and $\underline{c}$ in 3 -dimensions are said to be linearly independent if the only solution to the equation $\lambda_{1} a+\lambda_{2} b+\lambda_{3} c=Q$ is the trivial solution $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$. Determine whether or not each set of yectors is linearly independent. (a) $\underline{a}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \underline{b}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right], \underline{c}=\left[\begin{array}{c}0 \\ -1 \\ 1\end{array}\right]$ (b) $\underline{a}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \underline{b}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right], \underline{c}=\left[\begin{array}{c}0 \\ -1 \\ 2\end{array}\right]$
Solution
Step 1 ：Form the matrices using the given vectors and calculate their determinants:
Step 2 ：\(A_1 = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 0 & -1 & 1 \end{bmatrix}, \quad A_2 = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 0 & -1 & 2 \end{bmatrix}\)
Step 3 ：\(\det(A_1) = 0, \quad \det(A_2) = 1\)
Step 4 ：Since the determinant of \(A_1\) is 0, the vectors in (a) are linearly dependent.
Step 5 ：\(\boxed{\text{(a) Linearly Dependent}}\)
Step 6 ：Since the determinant of \(A_2\) is 1, the vectors in (b) are linearly independent.
Step 7 ：\(\boxed{\text{(b) Linearly Independent}}\)
