Chapter 4: Q33E (page 164)

Explain why two functions are linearly dependent on an intervalIif and only if there exist constants c₁and c₂,not both zero, such thatfor $c_{1} y_{1} (t) + c_{2} y_{2} (t) = 0$ alltinI.

Short Answer

Expert verified

The functions are linearly dependent.

Step by step solution

Apply the given values.

Here $c_{1} y_{1} (t) + c_{2} y_{2} (t) = 0$

If the y₁ and y₂ are linearly dependent then;

$y_{1} (t) = C_{1} y_{2} (t)$

Find linearly dependent functions

Now,

$y_{1} (t) - C_{1} y_{2} (t) = 0$

Let $C_{1} = - \frac{c_{2}}{c_{1}}$ then

$y_{1} (t) + \frac{c_{2}}{c_{1}} y_{2} (t) = 0 c_{1} y_{1} (t) + c_{2} y_{2} (t) = 0$

Let $c_{1} = 0 and c_{2} \neq 0$ then $y_{2} (t) = 0$ .

And if $c_{2} = 0 and c_{1} \neq 0$ then $y_{1} (t) = 0$

So, if both are not zero then both are linearly dependent.

Therefore, functions are linearly dependent.

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Explain why two functions are linearly dependent on an intervalIif and only if there exist constants c₁and c₂,not both zero, such thatfor $c_{1} y_{1} (t) + c_{2} y_{2} (t) = 0$ alltinI.

Short Answer

Step by step solution

Apply the given values.

Find linearly dependent functions

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Explain why two functions are linearly dependent on an intervalIif and only if there exist constants c1and c2,not both zero, such thatforc1y1(t)+c2y2(t)=0 alltinI.

Short Answer

Step by step solution

Apply the given values.

Find linearly dependent functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Decision Maths

Calculus

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Explain why two functions are linearly dependent on an intervalIif and only if there exist constants c₁and c₂,not both zero, such thatfor $c_{1} y_{1} (t) + c_{2} y_{2} (t) = 0$ alltinI.