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In: Advanced Math

Question 1 A) Show that the functions y1(t) = 1 + t 2 ; y2(t) =...

Question 1

A) Show that the functions y1(t) = 1 + t 2 ; y2(t) = 1 − t 2 are linearly independent directly from the definition of linear independence.

B)Find three functions y1(t), y2(t), y3(t) such that any two of them are linearly independent but three of them are not linearly independent.

Solutions

Expert Solution

a) Suppose that are such that . Then

implies

Second equation gives , and first equation gives

Thus, we have proved that if and only if . Thus, are linearly independent.

b) Let us consider as above, and . We know that are linearly independent by part a).

Proof of the linear independence of :

Suppose that are such that . Then

implies

Solving, we get

Thus, we have proved that if and only if . Thus, are linearly independent.

Proof of the linear independence of :

Suppose that are such that . Then

implies

Solving, we get

Thus, we have proved that if and only if . Thus, are linearly independent.

However, are not linearly independent because

Colby Messinger answered 2 years ago
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