Answered: X1(t) : X2(t)

Calculus: Early Transcendentals

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

Question

show that the given vector functions are linearly independent on (−∞,∞).

Expert Solution

Step 1

To show that the given vector functions are linearly independent

$x_{1} (t) = [\begin{matrix} e^{t} \\ - e^{t} \end{matrix}], x_{2} (t) = [\begin{matrix} e^{t} \\ e^{t} \end{matrix}]$

Step 2

let $a, b \in ℝ$ such that

$\begin{array}{rcl} a x_{1} + b x_{2} & = & 0 \\ a [\begin{matrix} e^{t} \\ - e^{t} \end{matrix}] + b [\begin{matrix} e^{t} \\ e^{t} \end{matrix}] & = & [\begin{matrix} 0 \\ 0 \end{matrix}] \\ a e^{t} + b e^{t} & = & 0 \\ e^{t} (a + b) & = & 0 \\ a + b & = & 0 (1) \\ - a e^{t} + b e^{t} & = & 0 \\ e^{t} (- a + b) & = & 0 \\ - a + b & = & 0 (2) \end{array}$

Solving (1) and (2)

$a + b = 0 - a + b = 0 2 b = 0 b = 0 a = 0$

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