The following definition is discussed in advanced mathematics courses.

Anish Buchanan 2021-10-05 Answered

The following definition is discussed in advanced mathematics courses. f(x)={0if xx is a rational number 1if xx is an irrational number
Evaluate f(34), 2, and f(π).

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ensojadasH
Answered 2021-10-06 Author has 100 answers

Since x=34 is a rational number (can be written as a ratio of two integers), then: f34=0
Since x=2 is an irrational number, then: f(2)=1
Since x=π is an irrational number, then: f(π)=1

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1a. 

 

Show from first principles, i.e., by using the definition of linear independence,
that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrix
A with associated eigenvector v = u + iw, then the two real solutions
Y(t) = eat(u cos bt − wsin bt)
and
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX.

 

1b. 

Use (a) to solve the system (see image)
 

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