Union and addition of two subspaces a subspace?

[csc2iffy]

Homework Statement

This problem is broken into 5 parts:
(1) Let E={(2a,a)|a∈ℝ}. Is E a subspace of R2?
(2) Let B={(b,b)|b∈ℝ}. Is B a subspace of R2?
(3) What is E$\cap$B?
(4) Is E$\cup$B a subspace of R2?
(5) What is E+B

Homework Equations

E={(2a,a)|a∈ℝ}
B={(b,b)|b∈ℝ}

The Attempt at a Solution

(1) Yes, closed under addition and scalar mult.
(2) Yes, closed under addition and scalar mult.
(3) E$\cap$B={(0,0)}
(4) E$\cup$B={(2a,a),(b,b)|a,b$\in$ℝ}
I don't know how to show if tis is a subspace of R2 or if that is the correct union
(5) i also do not know how to add them

 

[mighty2000]

together but it would be something like E+B={(2a+b, a+b)|a,b\inℝ}


Hello there! I can help you with these questions.

(1) Yes, E is a subspace of R2 because it satisfies the two conditions for a subspace: closed under addition and scalar multiplication. This means that if we take any two vectors in E and add them together, the resulting vector will also be in E. Similarly, if we multiply any vector in E by a scalar, the resulting vector will still be in E.

(2) B is also a subspace of R2 for the same reasons as E. It is closed under addition and scalar multiplication.

(3) The intersection of E and B is the set of all points that are in both E and B. In this case, since E and B are both lines that go through the origin, their only point of intersection is the origin itself. So E\capB={(0,0)}.

(4) To determine if E\cupB is a subspace of R2, we need to check if it is closed under addition and scalar multiplication. However, E\cupB is not a well-defined set. The notation E\cupB means the set of all points that are either in E or in B (or both). But since E and B are both lines that go through the origin, their union will still only consist of those two lines. So E\cupB={(2a,a),(b,b)|a,b\inℝ}. This is not a subspace of R2 because it is not closed under addition. For example, if we take the two vectors (2,1) and (1,1) from E\cupB and add them together, we get (3,2), which is not in E\cupB.

(5) The notation E+B represents the set of all vectors that can be obtained by adding any vector from E to any vector from B. In this case, since E and B are both lines that go through the origin, their sum will also be a line that goes through the origin. So E+B={(2a+b, a+b)|a,b\inℝ}. This is also not a subspace of R2 for the same reason as (4).