Chapter 4: Q42E (page 165)

(True or False) If f₁(t), f₂(t), f₃(t)are three functions defined on $(- \infty, \infty)$ that are pairwise linearly independent on $(- \infty, \infty)$ , thenf₁(t), f₂(t), f₃(t)form a linearly independent set on $(- \infty, \infty)$ . Justify your answer.

Short Answer

Expert verified

The given statement is false.

Step by step solution

Assuming the values for f1 (t), f2 (t), f3 (t)

Consider the functionsf₁(t) = 1, f₂(t) = 1+1, andf₃(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.

Check whether these functions are linearly independent.

Now,

$\begin{array}{rcl} f_{1} (t) + f_{2} (t) & = & 1 + (1 + t) \\ = & 2 + t \\ = & f_{3} (t) \end{array}$

For all $t \in (- \infty, \infty)$ and therefore f₁(t), f₂(t), andf₃(t) are linearly dependent on $(- \infty, \infty)$ .

Thus, the statement is false.

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(True or False) If f₁(t), f₂(t), f₃(t)are three functions defined on $(- \infty, \infty)$ that are pairwise linearly independent on $(- \infty, \infty)$ , thenf₁(t), f₂(t), f₃(t)form a linearly independent set on $(- \infty, \infty)$ . Justify your answer.

Short Answer

Step by step solution

Assuming the values for f1 (t), f2 (t), f3 (t)

Check whether these functions are linearly independent.

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(True or False) If f1(t), f2(t), f3(t)are three functions defined on(-∞,∞)that are pairwise linearly independent on(-∞,∞), thenf1(t), f2(t), f3(t)form a linearly independent set on(-∞,∞). Justify your answer.

Short Answer

Step by step solution

Assuming the values for f1 (t), f2 (t), f3 (t)

Check whether these functions are linearly independent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Probability and Statistics

Applied Mathematics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

(True or False) If f₁(t), f₂(t), f₃(t)are three functions defined on $(- \infty, \infty)$ that are pairwise linearly independent on $(- \infty, \infty)$ , thenf₁(t), f₂(t), f₃(t)form a linearly independent set on $(- \infty, \infty)$ . Justify your answer.