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Chapter 4: Q42E (page 165)

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

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

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

Consider the functionsf1(t) = 1, f2(t) = 1+1, andf3(t) = 2 defined on (-鈭�,鈭�). We have that the functions are pairwise linearly independent since none of them is a multiple of any of the other two.

02

Check whether these functions are linearly independent.

Now,

f1(t)+f2(t)=1+(1+t)=2+t=f3(t)

For allt鈭�(-鈭�,鈭�) and therefore f1(t), f2(t), andf3(t) are linearly dependent on(-鈭�,鈭�).

Thus, the statement is false.

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