Q42E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

(True or False) If f₁(t), f₂(t), f₃(t) are three functions defined on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$ that are pairwise linearly independent on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$, then f₁(t), f₂(t), f₃(t) form a linearly independent set on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$. Justify your answer.

Step-by-Step Solution

Answer

The given statement is false.

Step 1: Assuming the values for f 1 (t), f 2 (t), f 3 (t)

Consider the functions f₁ (t) = 1, f₂ (t) = 1+1, and f₃ (t) = 2 defined on $(-\infty ,\infty )$. We have that the functions are pairwise linearly independent since none of them is a multiple of any of the other two.

Step 2: Check whether these functions are linearly independent.

Now,

$\begin{array}{rcl}{f}_1\left(t\right)+f_2\left(t\right)& =& 1+(1+t)\\ & =& 2+t\\ & =& f_3\left(t\right)\end{array}$

For all $t\in (-\infty ,\infty )$ and therefore f₁ (t), f₂ (t), and f₃ (t) are linearly dependent on$(-\infty ,\infty )$.

Thus, the statement is false.