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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 thatforc1y1(t)+c2y2(t)=0 alltinI.

Short Answer

Expert verified

The functions are linearly dependent.

Step by step solution

01

Apply the given values.

Here c1y1(t)+c2y2(t)=0

If the y1 and y2 are linearly dependent then;

y1(t)=C1y2(t)

02

Find linearly dependent functions

Now,

y1(t)-C1y2(t)=0

LetC1=-c2c1 then

y1(t)+c2c1y2(t)=0c1y1(t)+c2y2(t)=0

Letc1=0鈥�鈥�and鈥�鈥�c2鈮�0 then y2(t)=0.

And ifc2=0鈥�鈥奱苍诲鈥�鈥�c1鈮�0 theny1t=0

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

Therefore, functions are linearly dependent.

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