Recent content by hkus10

Let use an example to illustrate source, sink, and node For example, let assume the equilibrium points are y = -3 and y = 2. dy/dt < 0 fir -3 < y < 2, and dy/dt > 0 for y < -3 and y > 2. Given this information, y = -3 is a sink and y = 2 is a source. Node just mean if the left hand and the...

Challenge! Use Taylor series expansions to prove first-order Differential Equation Suppose dy/dt = f(y) has an equilibrium point at y = y0 and a) f'(y0) = 0, f''(y0) = 0, and f'''(y0) > 0: Is yo a source, a sink, or a node? b) f'(y0) = 0, f''(y0) = 0, and f'''(y0) < 0: Is yo a source, a sink...

1) How to justify if there is a tie for the minimum b-ratio at some iteration of the phase II simplex algorithm, then the next basic feasible solution is degenerate. I have no idea how to justify it. Please give me some direction 2) Max. z = transpose of C * the vector x s.t...

Let P belongs to Mnn be a nonsingular matrix and Let L:Mnn>>>Mnn be given by L(A) = P^-1AP for all A in Mnn. Prove that L is an invertible linear operator. I have no clues how to start this question. What do I need to prove for this question? and why

Yes, this is a typo. It seems to me that if the eigenvalue is 0, L is not invertible. However, I cannot conceptualize it. Could you please explain it ? Thanks

a) This is what I get for part(a)let n=lambda. Since r is an eigenvalue of L, Lx=nx. Since the transformation is invertible, (L^-1)Lx=(L^-1)nx. ==> Ix=r(L^-1)x, where I=indentity matrix At this point, I want to divide both sides by r. However, how can I be sure r is not equal to zero? Thanks

Let L : V>>>V be an invertible linear operator and let lambda be an eigenvalue of L with associated eigenvector x. a) Show that 1/lambda is an eigenvalue of L^-1 with associated eigenvector x. For this question, the things I know are that L is onto and one to one. Therefore, how to prove this...

1) (x-10)[(x+4)(x+1) - 24] - 3[(-11x - 11) + 24] + 8[-21 + 3x] what I get is (x-10)(x^2+5x-20) + 57x-207 The reason that I do not combine them is because I think it is much more difficult to deal with x^3? What should I do here?

push!

For 2) Is the answer just simply 2 + 5? For 3) what do you mean by a linear map? Also, I have an additional question which is If L:V>>>W is a linear transformation of a vector space V into a vector space W and dim V = dim W, then prove that if L is one-to-one, then it is onto. I get a...

1) Let L:R3 >>>R3 be defined by L([1 0 0]) = [1 2 3], L([0 1 0]) = [0 1 1], L([0 0 1]) = [1 1 0] How to prove that L is invertible? I have the idea of one-to-one and onto, but I do not know how to apply them to this proof. 2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2], [3 1...

Thanks

Since L(at^2 + bt + c) = 2at^3 + bt^2 No matter what value of t and 1, 2a^3 + bt^2 should always give me 0 vector. So, I have a question why Ker(L) does not have t as a basis? Another question is dim Ker(L) + dim range(L) = dim (p3) by thm. since the dim range(L) = 2 and dim (p3) = 4, why dim...

is the basis for ker(L) {t, 1} and the basis for range(L) {t^3, t^2}?

Let L:p2 >>> p3 be the linear transformation defined by L(p(t)) = t^2 p'(t). (a) Find a basis for and the dimension of ker(L). (b) Find a basis for and the dimension of range(L). The hint that I get is to begin by finding an explicit formula for L by determining L(at^2 + bt + c). Does...