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