# Show that a line in R2 is a subspace problem

• negation
In summary, to show that a line in R2 is a subspace if and only if it passes through the origin (0,0), we need to prove two statements: 1. If a line passes through the origin, it is a subspace. 2. If a line is a subspace, it must pass through the origin.
negation

## 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?

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.

maajdl said:
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:
negation said:

## 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.
negation said:
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.
negation said:
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.

## 1. What is a subspace in R2?

A subspace in R2 is a subset of the two-dimensional Cartesian plane that satisfies certain conditions. These conditions include containing the origin (0,0), being closed under addition and scalar multiplication, and being closed under the operations of addition and multiplication.

## 2. How do you show that a line in R2 is a subspace?

To show that a line in R2 is a subspace, we must prove that it satisfies the conditions of a subspace. This can be done by showing that the line contains the origin, and that it is closed under addition and scalar multiplication. Additionally, we must show that the operations of addition and scalar multiplication on the line produce points that are also on the line.

## 3. What is the equation of a line in R2?

The equation of a line in R2 is typically written in slope-intercept form, y = mx + b, where m represents the slope of the line and b represents the y-intercept. In order for a line to be a subspace in R2, the y-intercept must be 0, and the slope must be a real number.

## 4. Can a line in R2 be a subspace if it does not pass through the origin?

No, a line in R2 cannot be a subspace if it does not pass through the origin. This is because one of the conditions for a subspace is that it must contain the origin (0,0). If a line does not pass through the origin, it cannot be closed under the operations of addition and scalar multiplication, and therefore cannot be a subspace.

## 5. What is the importance of showing that a line in R2 is a subspace?

Showing that a line in R2 is a subspace is important because it helps us understand the properties and behaviors of lines in two-dimensional space. It also allows us to use the tools and techniques of linear algebra to solve problems and make predictions about these lines.

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