Results 1 to 5 of 5

Math Help - Which subsets of R^3 are subspaces?

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    22

    Which subsets of R^3 are subspaces?

    SUBSPACES !

    Hi, I don't understand subspaces! I came across this question in homework tasks, and I'm guessing it's very important to the course?!

    Which of the following subsets of R^3 are subspaces? (Justify answers)

    (a) A = {(s, 0, 2s) | s Є R }
    (b) B = {(x, y, z) | y ≥ z}
    (c) C = {(t+1, t, 3t) | t Є R }
    (d) D = {(x, y, z) | 4x + y - z = 0}


    Help? Please? :-)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by iExcavate View Post
    SUBSPACES !

    Hi, I don't understand subspaces! I came across this question in homework tasks, and I'm guessing it's very important to the course?!

    Which of the following subsets of R^3 are subspaces? (Justify answers)

    (a) A = {(s, 0, 2s) | s Є R }
    (b) B = {(x, y, z) | y ≥ z}
    (c) C = {(t+1, t, 3t) | t Є R }
    (d) D = {(x, y, z) | 4x + y - z = 0}


    Help? Please? :-)
    A subspace is a vector space which is contained in the bigger vector space.

    If you look at your notes you will see that you only have two things to prove - closure under addition and closure under scalar multiplication. This is because we get the other axioms for free from the big space.

    That is, if you are wanting to prove that for U \subset V that U is a subspace of V you need to check that \alpha u \in U for all \alpha in the field, u \in U, and that for all u_1, u_2 \in U, u_1+u_2 \in U.

    I believe two of the four are subspaces, the other two are not closed under scalar multiplication.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,794
    Thanks
    1532
    Quote Originally Posted by iExcavate View Post
    SUBSPACES !

    Hi, I don't understand subspaces! I came across this question in homework tasks, and I'm guessing it's very important to the course?!

    Which of the following subsets of R^3 are subspaces? (Justify answers)
    To be a subspace, we must have closure under addtion and scalar multiplication- and we can put those together- A is a subspace if and only if, for all u and v in A and scalars a and b, au+ bv is also in A.

    (a) A = {(s, 0, 2s) | s Є R }
    Let u= (s, 0, 2s) and v= (t, 0, 2t). au+ bv= (as, 0, 2as)+ (bt, 0, 2bt)= (as+ bt, 0, 2(as+ bt)). Is that also in A?

    (b) B = {(x, y, z) | y ≥ z}
    Is -1(x, y, z) also in B?

    (c) C = {(t+1, t, 3t) | t Є R }
    Is 0(t+1, t, 3t) in C? (The 0 vector, of course, is in every subspace.)

    (d) D = {(x, y, z) | 4x + y - z = 0}
    Let u= (x, y, z) and v= (i, j, k) where 4x+ y- z= 0 and 4i+ j- k= 0 so that they are in D. au+ bv= (ax, ay, az)+ (bi, bj, bk)= (ax+ bi, ay+ bj, az+ bk). For that to be in D, we must have 4(ax+ bi)+ (ay+ bj)- (az+ bk)= 0. Rewrite that as (4ax+ ay- az)+ (4bi+ bj- bk= a(4x+ y- z)+ b(4i+ j- k).

    Help? Please? :-)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2009
    Posts
    22
    Quote Originally Posted by HallsofIvy View Post

    Is -1(x, y, z) also in B?

    I don't know? Is it? Wouldn't it just be (-x, -y, -z) with the rule saying that y is greater than or equal to z?

    Also, I found A is a subspace, B not sure yet, C isn't a subspace, and I'm guessing D is a subspace?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by iExcavate View Post
    I don't know? Is it? Wouldn't it just be (-x, -y, -z) with the rule saying that y is greater than or equal to z?

    Also, I found A is a subspace, B not sure yet, C isn't a subspace, and I'm guessing D is a subspace?
    If y \geq z then is -y \geq -z? Take an example, for instance 4 \geq 3. Is -4 \geq -3?

    Part D does give you a subspace. What you need to do is check that the equation holds when you multiply by a scalar and when you add two vectors together. That is, if 4x_1 + y_1 - z_1 = 0 and if 4x_2 + y_2 - z_2 = 0 then does

    4\alpha x_2 + \alpha y_2 + \alpha z_2 = 0 hold? (As (\alpha x_1, \alpha y_1, \alpha z_1) must be of the form required).

    4(x_1 + x_2) + (y_1 + y_2) - (z_1 + z_2) = 0 hold? (As (x_1 + x_2, y_1 + y_2, z_1 + z_2) must be of the form required).

    Quote Originally Posted by HallsofIvy View Post
    To be a subspace, we must have closure under addtion and scalar multiplication- and we can put those together- A is a subspace if and only if, for all u and v in A and scalars a and b, au+ bv is also in A.


    Let u= (s, 0, 2s) and v= (t, 0, 2t). au+ bv= (as, 0, 2as)+ (bt, 0, 2bt)= (as+ bt, 0, 2(as+ bt)). Is that also in A?


    Is -1(x, y, z) also in B?


    Is 0(t+1, t, 3t) in C? (The 0 vector, of course, is in every subspace.)


    Let u= (x, y, z) and v= (i, j, k) where 4x+ y- z= 0 and 4i+ j- k= 0 so that they are in D. au+ bv= (ax, ay, az)+ (bi, bj, bk)= (ax+ bi, ay+ bj, az+ bk). For that to be in D, we must have 4(ax+ bi)+ (ay+ bj)- (az+ bk)= 0. Rewrite that as (4ax+ ay- az)+ (4bi+ bj- bk= a(4x+ y- z)+ b(4i+ j- k).
    What was wrong with my first post, and whatever happened to letting the OP think for themselves?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Which of the following subsets are subspaces?
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 12th 2011, 03:47 AM
  2. Subsets that are subspaces of P2
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 17th 2011, 08:02 PM
  3. Replies: 3
    Last Post: May 3rd 2009, 11:35 PM
  4. Verifying Subsets as Subspaces through scalar vector
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: April 25th 2009, 11:38 PM
  5. Subsets and subspaces, perpendicular sets!
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 6th 2008, 11:51 PM

Search Tags


/mathhelpforum @mathhelpforum