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Show that a line in R2 is a subspace problem

  1. Mar 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that a line in R2 is a subspace if and only if it passes through the origin (0,0)

    3. The attempt at a solution

    S={(x,y)| (x,y) =(0,0)}

    S = {(x,y)|x=y}

    Am I setting up the problem correctly?
  2. jcsd
  3. Mar 5, 2014 #2


    User Avatar
    Gold Member

    No, you are not setting up the problem, you are guessing anything.
    First, you need to understand the meaning of each word in the problem.
    What are the meaning of:

    - a line in R2, what is that?
    - what is a subspace, very important: you must be able to explain what is a subspace
    - since subspace is a space, you need to know the meaning of a space too
    - finally what is the meaning of a line passing through the origin

    How are you used to represent a line? Star with that.
  4. Mar 5, 2014 #3
    I wasn't guessing. I never make guesses to a problem. The learning curve has been very steep for this unit and the lectures isn't quite helpful. I want to cement my understanding of vector space and subspace by THIS week.

    I have one fundamental problem in my understanding of vector space.
    E.g., S = {(x,y)|. . . }
    What does (x,y) stands for?

    As to your question, and I shall do to my best of my current understanding to answer:

    -A line in R2 implies a plane in a 2-dimension.

    -A subspace is a subset of a vector space. Let's suppose S is a collection of vectors. For S to qualify as a subspace, the vectors as member vector of S must:
    1) be closed under addition such that if u and v are member vectors of S, u + v must equally be member vectors of S.
    2) be closed under scalar multiplication such that if u is a member vector of S, and k is a scalar of the field line, then u. k must be a member vector of S.
    3) the zero-vector must be a vector member of S.

    -A line passing through an origin is a plane/ line cutting through the point (0,0)

    - A line can be represented by the equation y = mx + c
    Last edited: Mar 5, 2014
  5. Mar 5, 2014 #4


    Staff: Mentor

    Here, S is a set consisting of a single point - the origin.
    S is the line whose equation is y = x. There are many (an infinite number) other lines in the plane that pass through the origin.
    No, not at all. You need to prove two things:
    1. If a line in R2 passes through the origin, the line is a subspace of R2.
    2. If a line is a subspace of R2, it must pass through the origin.
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