Show that a line in R2 is a subspace problem

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  • #1
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Homework Statement


Show that a line in R2 is a subspace if and only if it passes through the origin (0,0)



The Attempt at a Solution



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

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

Am I setting up the problem correctly?
 

Answers and Replies

  • #2
maajdl
Gold Member
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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.
 
  • #3
818
0
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.

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:
  • #4
35,132
6,878

Homework Statement


Show that a line in R2 is a subspace if and only if it passes through the origin (0,0)



The Attempt at a Solution



S={(x,y)| (x,y) =(0,0)}
Here, S is a set consisting of a single point - the origin.
Or
S = {(x,y)|x=y}
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.
Am I setting up the problem correctly?

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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