Results 1 to 3 of 3

Math Help - prove subgroup of GL2(R)

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    21

    prove subgroup of GL2(R)

    Hi, I think I have this correct but I need someone to confirm my working.

    Let G be the set of matrices of the form:
    [a b]
    [0 1]
    for a,b \in \mathbb{R}
    Prove that G is a subgroup of GL2(R).
    1. if x,y \in G, then xy \in G
    Let x=[a b : 0 1] and y =[c d : 0 1]
    Then
    xy=[ac ad+b : 0 1]
    \implies xy \in G

    2. If a=1, b=0, then I = [1 0 : 0 1]
    \implies I \in G

    3. if x \in G, then x^{-1} \in G.
    x=[a b : 0 1], then x^{-1} = [1 -b : 0 a]
    thus, x^{-1} \in G.

    Thus, G is a subgroup of GL2(R).

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: prove subgroup of GL2(R)

    Quote Originally Posted by shelford View Post
    Let G be the set of matrices of the form:
    [a b]
    [0 1]
    for a,b \in \mathbb{R}
    Prove that G is a subgroup of GL2(R).
    It should be ... with a\neq 0

    1. if x,y \in G, then xy \in G
    Let x=[a b : 0 1] and y =[c d : 0 1]
    Then
    xy=[ac ad+b : 0 1]
    \implies xy \in G
    ... because ac\neq 0

    2. If a=1, b=0, then I = [1 0 : 0 1]
    \implies I \in G
    ... i.e. the identity element of \textrm{GL}_2(\mathbb{R}) belongs to G .

    3. if x \in G, then x^{-1} \in G.
    x=[a b : 0 1], then x^{-1} = [1 -b : 0 a]
    thus, x^{-1} \in G.
    x^{-1}=\dfrac{1}{a}\begin{bmatrix}1&-b\\0&a \end{bmatrix} ... i.e. x^{-1}\in G

    P.S. If you use the characterization theorem of subgroups you only need to prove G\neq \emptyset and xy^{-1} for all x,y\in G .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2011
    Posts
    21

    Re: prove subgroup of GL2(R)

    Thanks FernandoRevilla, I understand it now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove C is a subgroup of G...
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 29th 2010, 12:55 AM
  2. Prove that for any a E G, Ca is a subgroup of G.
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 4th 2010, 09:31 AM
  3. Prove C(a) is a subgroup of G
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 24th 2010, 09:20 AM
  4. Prove C(H) is a subgroup of G
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 28th 2008, 10:08 AM
  5. prove for subgroup
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 20th 2008, 08:15 PM

Search Tags


/mathhelpforum @mathhelpforum