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![Problem A. Consider the linear transformation
T: R₁ [x] → R₁ [x]
given by
T(a + bx) = (a + b) + ax
(A1) Is T cyclic?
(A2) Is T irreducible?
(A3) Is T indecomposable?
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae19e8bd-5915-473c-8a03-9ac053dffb7e%2F98164fac-4685-4368-8433-34c1b6ff160b%2F9mnt3ok_processed.jpeg&w=3840&q=75)
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- Consider the linear transformation T : R2[x] → R2[x] given by T(a + bx + cx2 ) = (a − b − 2c) + (b + 2c)x + (b + 2c)x2 1) Is T cyclic? 2) Is T irreducible? 3) Is T indecomposable?Find the kernel for the transformation T(x) = Ax if? = [1 2 0 1 −1] |2 1 3 1 0| |−1 0 −2 0 1| [0 0 0 2 8]Two interacting populations of hares and foxes can be modeled by the recursive equations h(t + 1) = 4h(t) − 2 f (t) f(t+1)=h(t)+ f(t). For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f (t). a. h(0)= f(0)=100 b. h(0)=200, f(0)=100 c. h(0)=600, f(0)=500
- the relationship between aphids, A, ( prey) and ladybugs, L, (preditor) can be described as follows: dA/dt=2A=0.01AL dL/dt=-0.5L+0.0001AL a) find the two critical points of the predator-prey equations b) use the chain rule to write dL.dA in terms of L and A c) suppose that at time t=0, there are 1000 aphids and 200 ladybugs, use the Bluffton university slope field generator to graph the slope for the system and the solution curve. let 0 be less than or equal to A who is less than or equal to 15000 and 0 is less than or equal to L which is less than or equal to 400.If {~v1,··· ,~vr} is linearly independent and T is a one to one linear transformation, show that {T~v1,··· ,T~vr} is also linearly independent. Give an example which shows that if T is only linear, it can happen that, although {~v1,··· ,~vr} is linearly independent, {T~v1,··· ,T~vr} is not. In fact, show that it can happen that each of the T~vj equals 0.Let L : P1 → P1 be a linear transformation defined by L(t − 1) = t + 2 and L(t + 1) = 2t + 1. (a) What is L(5t + 1)? (b) What is L(at + b)?
- Use Laplace transforms to solve the initial value problem x'=2x+y, y'=6x+3y; x(0)=1,y(0)=-2Suppose T : P2 → R^2 is a linear transformation. If B = {1, x, x^2} andD = {(1, 1),(0, 1)}, find the action of T given MDB(T) = 1 2 −1 −1 0 1 .d) Prove that the transformation z = y^1−n reduces the equation dy dx+P(x)y = Q(x)yn to a linear equation in z and x. Hence, solve the initial value problem dy/dx + y/x=x/y^3