Question

The given vectors are solutions of a system X’ = AX. Determine whether the vectors form a fundamental set on the interval $(-\infty ,\infty )$.

${\mathit{X}}_{\mathbf{1}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{-}\mathbf{2}\\ \mathbf{4}\end{array}\mathbf{\right)}\mathbf{+}\mathit{t}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\\ \mathbf{2}\end{array}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{}{\mathit{X}}_{\mathbf{2}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{-}\mathbf{2}\\ \mathbf{4}\end{array}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{}{\mathit{X}}_{\mathbf{3}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{3}\\ \mathbf{-}\mathbf{6}\\ \mathbf{12}\end{array}\mathbf{\right)}\mathbf{+}\mathit{t}\mathbf{\left(}\begin{array}{c}\mathbf{2}\\ \mathbf{4}\\ \mathbf{4}\end{array}\mathbf{\right)}$

Short answer

The solution of given vector systemis $W\left(X_1,X_2,X_3\right)=0$.

Step-by-step solution

Fundamental set of solution;

Any set ${\mathit{X}}_{\mathbf{1}}\mathbf{,}{\mathit{X}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{X}}_{\mathbf{n}}$ be a fundamental set of solution of the homogeneous system on an interval l is said to be a fundamental set of solution on interval.

Evaluating the given vector:

The given vector is

$X_1=\left(\begin{array}{c}1\\ -2\\ 4\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ 2\end{array}\right),X_2=\left(\begin{array}{c}1\\ -2\\ 4\end{array}\right),X_3=\left(\begin{array}{c}3\\ -6\\ 12\end{array}\right)+t\left(\begin{array}{c}2\\ 4\\ 4\end{array}\right)$

Now use of determinant then,

$$\begin{gathered} W\left(X_1,X_2,X_3\right)=\left|\begin{array}{ccc}1+t& 1& 3+2t\\ -2+2t& -2& -6+4t\\ 4+2t& 4& 12+4t\end{array}\right| \\ =\left(1+t\right)\left|\begin{array}{cc}-2& -6+4t\\ 4& 12+4t\end{array}\right|-\left|\begin{array}{cc}-2+2t& -6+4t\\ 4+2t& 12+4t\end{array}\right|+\left(3+2t\right)\left|\begin{array}{cc}-2+2t& -2\\ 4+2t& 4\end{array}\right| \\ =\left(1+t\right)\left(-24t\right)-12t+\left(3+2t\right)\left(12t\right) \\ =-24t-24t^2-12t+36t+24t^2 \\ =0 \end{gathered}$$

We see that $X_1,X_2,X_3$are linearly dependent on the interval $\left(\infty ,-\infty \right)$, and hence they do not form a fundamental set on interval$\left(-\infty ,\infty \right)$

Hence, the final answer is$W\left(X_1,X_2,X_3\right)=0$