Results 1 to 3 of 3

Math Help - Linear maps

  1. #1
    Junior Member
    Joined
    Feb 2006
    From
    Victoria, Australia
    Posts
    36

    Linear maps

    i have 2 questions im finding hard to try and answer.

    Let T1 : U -> V and T2 : V -> W be linear maps.
    Show that the map T2 T1 : U ->W is linear.



    and


    A matrix which satisfies A^T = -A is called skew-symmetric. Show that if A is an n*n skew-symmetric matrix where n is an odd integer, then A has no inverse. (Hint: use the definition A^T = -A to find det A, ) but be careful!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hi
    Quote Originally Posted by sterps View Post
    i have 2 questions im finding hard to try and answer.

    Let T1 : U -> V and T2 : V -> W be linear maps.
    Show that the map T2 T1 : U ->W is linear.
    Let \lambda \in \mathbb{K},\,u_1,\,u_2\in U, you need to show that T_2\circ T_1(\lambda u_1+u_2)=\lambda T_2\circ T_1(u_1)+T_2\circ T_1(u_2) acknowledged that T_1 and T_2 are both linear maps.
    A matrix which satisfies A^T = -A is called skew-symmetric. Show that if A is an n*n skew-symmetric matrix where n is an odd integer, then A has no inverse. (Hint: use the definition A^T = -A to find det A, ) but be careful!
    You got the hint... (and I even think that the solution has been posted in this forum few days ago)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by sterps View Post
    i have 2 questions im finding hard to try and answer.

    Let T1 : U -> V and T2 : V -> W be linear maps.
    Show that the map T2 T1 : U ->W is linear.
    and
    A matrix which satisfies A^T = -A is called skew-symmetric. Show that if A is an n*n skew-symmetric matrix where n is an odd integer, then A has no inverse. (Hint: use the definition A^T = -A to find det A, ) but be careful!
    1)If T_1 and T_2 are linear maps

    Let u,v \in U, then T_1(u+v) = T_1(u) + T_1(v) \in V
    Since T_1(u),T_1(v) \in V, then T_2(T_1(u+v))) = T_2(T_1(u) + T_1(v)) = T_2(T_1(u)) + T_2(T_1(v)) \in W

    Thus \forall u,v \in U, T_2(T_1(u+v))) = T_2(T_1(u)) + T_2(T_1(v))\in W

    Do similarly for scalar multiplication, thus proving that the map T2 T1 : U ->W is linear.

    2) Claim: det(A) = 0

    A^T = -A

     \Rightarrow det(A^T) = det(-A)

    \Rightarrow det(A) = (-1)^n \,\, det(A) [\because \forall A_{n \times n}, \,\, det(\lambda A) = \lambda ^n\,det(A) \text{and } det(A^T) = det(A)]

    \Rightarrow det(A) = -det(A)[\because \text{n = odd}]

    \Rightarrow det(A) = 0
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 10th 2010, 12:38 PM
  2. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 9th 2010, 04:42 AM
  3. Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 15th 2009, 10:27 AM
  4. Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 2nd 2009, 12:14 AM
  5. Linear maps. Proving its linear, and describing kernal.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 20th 2008, 01:46 AM

Search Tags


/mathhelpforum @mathhelpforum