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Math Help - Help with a few questions

  1. #1
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    Help with a few questions

    1) Which of the following are subspaces of R3

    a) T = {(x1, x2, x3) | x1x2x3 = 0}
    b) T = {(x1, x2, x3) | x1 - x3 = 0}
    c) T = {(x1, x2, x3) | x1 = 0}
    d) T = {(x1, x2, x3) | x1 = 1}

    2) Let U be a k-vector space, where k is any field. Let V be a subspace of U. Assume that dimk V = dimk U. Show that U = V

    3) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Za.

    4) Let T: U--->V be a linear map. Show that T(0u) = 0v

    5) Find an example of vector space V and two subspaces W ⊂ V and Z ⊂ V such that Z ∪ W is not a subspace?

    6) Show that T = {a + b√3 | a,b∈T } is a field (a subfield of R). Show that M = {a + b√3 | a,b∈ L} is not a field

    Any help would be appreciated
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Help with a few questions

    Those are too many questions in only one thread. Besides, you should show some work according to the rules of the forum. A little help, so other members can also answer you.

    Quote Originally Posted by MastersMath12 View Post
    1) Which of the following are subspaces of R3 a) T = {(x1, x2, x3) | x1x2x3 = 0}
    (1,1,0) and (0,0,1) belong to T but (1,1,0)+(0,0,1)=(1,1,1)\not\in T, so T is not a subspace of \mathbb{R}^3.
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  3. #3
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    Re: Help with a few questions

    I made a mistake. Those are actually the questions I had answered. I coppied those by accident. Sorry

    Sorry, I am new to the forum.

    1) Find two linear maps A,B: R2 ---> R2 such that AB ≠ BA

    I understand how to find the two linear maps, but I am still lost with respect to "such that AB ≠ BA"

    2) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Z2.

    For this one, I am just completely confused. I understand the idea of proving they are linearly independant but I am having trouble prooving if the same conclusion would be true over Z2.

    Any help would be greatly appreciated.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Help with a few questions

    Quote Originally Posted by MastersMath12 View Post
    1) Find two linear maps A,B: R2 ---> R2 such that AB ≠ BA .I understand how to find the two linear maps, but I am still lost with respect to "such that AB ≠ BA"
    Choose A(x_1,x_2)=(x_2,0),\;B(x_1,x_2)=(0,x_1). Verify that A\circ B\neq B\circ A.

    2) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Z2.
    It is not true over \mathbb{Z}_2=\{0,1\}. Counterexample: choose u=(1,0) and v=(0,1), then \{u,v\} are linearly independent, but u+v=(1,1) and u-v=(1,-1)=(1,1), so u + v and u-v are not linearly independent.
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