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)
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX.
Use (a) to solve the system (see image)
A car travels at 108 kph from A to B for t seconds, applies the brakes for 4 seconds between B and C to give the car a constant deceleration and a speed of 72 kph at C, and finally travels at this speed from C to D, also for t seconds. If the total horizontal distance from A to D is 3100 m, determine the distance between A and B.
In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of