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Everything posted by hkus10

  1. 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
  2. a) This is what I get let n=lambda. Since r is an eigenvalue of L, Lx=nx. Since the transformation is invertible, (L^-1)Lx=(L^-1)nx. ==> Ix=n(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
  3. What does the sentence want me to write down? The key is I do not understand the question.
  4. 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 question? b) state the analogous statement for matrices. What does "state" the analogous statement mean? Thanks
  5. 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?
  6. <font face="Arial">1) Let L:R3 >>>R3 be defined by<br> L([1 0 0]) = [1 2 3],<br> L([0 1 0]) = [0 1 1],<br> L([0 0 1]) = [1 1 0]<br><br> 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.<br><br> 2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2], [3 1 -1]} is a basis for range (L). How can I approach this problem?<br><br> 3) Let S = {v1,…,vn} be an ordered basis for vector space V . Let<br> L :V →R^n be given by L(v) = [v]S . Prove that L is an<br> isomorphism( prove that the linear transformation is one-to-one and onto)<br> What I know so far is that {[v1]s, [v2]s,...,[vn]s} is an ordered basis for R^n and v can be written in a unique way sych that v = a1v1+...+anvn = 0.<br> How can I go from there?<br><br> 4) 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.<br> I get a point that dim(W) = dim(range(L)). What I am trying to get to is W = range(L). Then, by definition, L is onto. My question is that how can I prove that dim(W) = dim(range(L)) >>> W = range(L) and by what Thm of defin? If not, how should I approach this question?<br><br> 5) Let L:R^n>>>R^m be a linear transformation defined by L(x) = Ax, for x in R^n. Then, L is onto if and only if rank A = m.<br> In this question, is the nullity of A equal to the nullity of L?<br><br> Thanks </font>
  7. 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 this hint mean let p(t) = at^2 + bt + c? Then, I find that t^2 p'(t) = 2at^3 + bt^2. is the basis for ker(L) {t, 1} and the basis for range(L) {t^3, t^2}? Thanks
  8. hkus10


    Let S = {w1, w2, ..., wn} be a set of n vectors in R^n and let A be nxn matix whoise columns are the elements of S. Prove that for all b belong in R^n, Ax = b is consistent if and only if b belongs span(S). My approach is: I use contrapositive method to prove both sides First, I prove that if Ax = b is consistent, the b belongs span(s). Assume that Ax = b is inconsistent. Let x be [x1 x2 ... xn] ***This is a vectical vector which means x1, x2, and xn lines up vectically since I cannot express it in this way. This means the last row of all vectors in S are zeros and the last row of b has nonzero integer. Then, b cannot be written as a linear combination of vectors in S since 0x1 + 0x2 + ... + 0xn = 0. Therefore, b does not belong span(S). For the other side: Assume b does not belong span(S). Then, b cannot be written as a linear combination of vectors in S. If the last row of b is an nonzero integer, then the last row of all vectors in S must be zeros so that b cannot be written as a linear combination of vectors in S. By the Matrix-Vector Product written in terms of columns, [v1 v2 ... vn][x] not equal to . Thus, Ax = b is inconsistent. My question is that this proof seems quite reasonably for me. However, am I really proving this question. If not, how to approach it instead? Thanks
  9. Let L:P1 >> P1 be a linear transformation for which we know that L(t + 1) = 2t + 3 and L(t - 1) = 3t -2 a) Find L(6t-4) I just want to check the way to calculate this question. Is L(6t - 4) equal to 6*3t - 4*2 = 18t - 8? if not, how to calculate it?
  10. 1) Let S be an ordered basis for n-dimensional vector space V. Show that if {w1, w2, ..., wk} is a linearly independent set of vectors in V, then {[w1]s, [w2]s,...,[wk]s} is a linearly independent set of vectors in R^n. What I got so far is w1 = a1V1 + a2V2 + ... + anVn so, [w1]s = [a1 a2 ... an] The same thing for w2, [w2]s and wk, [wk]s. My question how to go from there? 2) Let S and T be two ordered bases of an n-dimensional vector space V. Prove that the transition matrix from T - coordinates to S - coordinates is unique. That is, if A,B belong to Mnn both satisfy A[v]T = [V]S and B[V]T = [v]S for all v belong to V, then A = B. My approach for this question is that Let S = {v1, v2, vn} Let T = {w1, w2, wn} Av = a1v1+a2v2+...+anvn v = b1w1+b2w2+...+bnwn Aa1v1 + Aa2v2+ ... +Aanvn a1(Av1) + a2(Av2)+...+an(Avn) = b1w1+b2w2+...+bn(wn) Am I going the right direction? If no, how should I approach? If yes, how should I move from here?
  11. 1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV. 2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V. 3) How to prove that the subspace of R^3 are{0}, R^3 itself, and any line or plane passing through the origin. How to approach these three Questions? Thanks
  12. Let V be the vector space of all real-valued continuous functions. t, e^t, sin(t) are in V. Is t, e^t, sin(t) in V linearly independent? My answer is yes. However, how can I prove it which is that which do I have to show or can I just say the def of linear independent?
  13. Ap + A(su) + A(tv) = b Ap + s(Au) + t(Av) = b Ap + s(0) + t(0) = b Ap = b Is this correct?
  14. Suppose that the solution set to a linear system Ax = b is a plane in R^n with vector equation x = p + su + tv , s, t ∈ R . Prove that p is a solution to the nonhomogeneous system Ax = b , and that u and v are both solutions to the homogeneous system Ax = 0 . (Hint Try choices of s and t). Should I start from A(p + su + tv) = b? If yes, what should I do from here? If no, where should I start?
  15. 1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other. 2) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V. Prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S. For these two questions, I know I have to prove them in both directions because of "if and only of". However, how to approach this problem? what Thms or definition should I use to prove them? 3) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span S. For question 3, does "W be a subspace of V containing S" mean W contains S? If yes, what is the reason to show it?
  16. hkus10


    Let T be the set of all matrics of the form AB - BA, where A and B are nxn matrics. Show that span T is not Mnn. 1) does "span T is not Mnn" mean that Mnn does not span T? Thanks
  17. hkus10


    Determine whether the given vector A in 2x2 matrix belongs to span{A1, A2, A3}, where A1 = [1 -1 0 3] A2 = [1 1 0 2] A3 = [2 2 -1 1] A = [5 1 -1 9]. Since A1, A2, A3 are not a nx1 matrices, I cannot put this into reduced echelon form? Therefore, what can I do to solve this problem? Thanks
  18. hkus10


    1) Let x0 be a fixed vector in a vector space V. Show that the set W consisting of all scalar multiples cx0 of x0 is a subspace of V. What techniques should I use to prove this? 2a) Show that a line lo through the origin of R^n is a subspace of R^n. 2b) show that a line l in R^n not passing through the origin is not a subspace of R^n. What techniques and direction should I use to solve these problems? Thanks
  19. Is this the answer? aW_1 + bW_2 = [math] \begin{pmatrix}a & b & a+b\\ a & 0 & 0\end{pmatrix} [/math] where a, b can be any real number.
  20. What I get is [math]\begin{pmatrix}a & b & 2c\\ a & 0 & 0\end{pmatrix}[/math] which is not [math]\begin{pmatrix}a & b & c\\ a & 0 & 0\end{pmatrix}[/math]
  21. 1.) The set W of all 2x3 matrices of the form a b c a 0 0 where c = a + b, is a subspace of M23 (Matrics 23). Show that every vector in W is a linear combination of W1 = 1 0 1 1 0 0 W2 = 0 1 1 0 0 0 Do I have to combine both W1 and W2 into one equation?
  22. 1) Find a vector equation for the plane in R3(3D) with scalar equation 2x − 3y + z = 5 . First,I find three points on the plane and then I used one point as a fixed point in order to find two vectors on the plane by using two other points. Then, I tried to test whether the two vectors are perpendicular to the normal vector of this plane by using cross product. However, I do not get the right normal vector by using my two vectors on the plane? My question is that whether my approach is wrong? If yes, what should I do?
  23. 1) If det(AB) = 0, is det(A) or det(B) = 0? Give reasons for your answer. Q1) First, cannot both det(A) or det(B) be 0? If it can, is this statement false. In any case, how can I prove that this is true for all statement since I only know how to find an example to show this is true, which cannot represent all the possibility. 2) Show that if A is singular and Ax = b, b is not equal to 0, has one solution, then it has infinitely many. Q2) How to approach this question? 3) Let A^2 = A. Prove that either A is singular or det(A) = 1. Q3) How can I approach this question?
  24. 1) Find an equation relating a, b, and c so that the linear system 2x+2y+3z = a 3x- y+5z = b x-3y+2z = c is consistent for any values of a, b, and c that satisfy that equation. what is the method to solve this problem? 2) In the following linear system, determine all values of a for which the resulting linear system has a) no solution; b) a unique solution; c) infinitely many solutions: x + y - z = 2 x + 2y + z = 3 x + y + (a^2 - 5)z = a For these two questions: Do I make this to be a reduced echelon form first? If yes, how to make it with some variables a, b, and c? If no, what is the right approach for this problem? Thanks
  25. 1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O. Can you show me the steps of solving this problem? Please!
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