How do I prove the subspace property for M and N in Linear Algebra?

In summary, the homework statement is trying to prove that M and N are subspace of V, but is having difficulty doing so. M and N are also said to be the intersection of M and N, but this is not completely clear. It is unclear what the first steps are to show that M and N intersect at the zero vector.
  • #1
1LastTry
64
0

Homework Statement



Let V = V1 + V2, where V1 and V2 are vector spaces. Define M ={(x1, 0vector2): x1 in V1}

and N = {(0vector1, x2) : x2 in V2
0vector 1 is the 0v of V1 and 0vector is the 0v of V2 and 0v is 0 vector of V

a) prove hat both M and N are subspace of V
b) show that M n N = {0v}
c) show that M+N=V

Homework Equations





The Attempt at a Solution


I am not clear about what M intersection N is
is it that the intersection of M and N is the 0 vector? If so what are the first steps to show this?

as for a)

do uprove using cx1 + x2, where Yi = (Yi, 0v2) Yj = (Yj,0v2)
and so on...?
 
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  • #2
For the intersection, (x,y) is in M and N if x=? and y=?.

For doing a, your notation is terrible so it's impossible to say whether you are making a reasonable statement. I say your notation is terrible because you start with vectors x1 and x2, and then you have vectors Yi and Yj coming from nowhere whose definitions depends on themselves.
 
  • #3
let y1 = (m1, 0v2) be in M and let y2 = (0v1, m2) also be in M

so prove that cy1 + y2 is also in M?
 
  • #4
Yes, except you should prove it for y2 = (m2, ov2)!
 
  • #5
Sorry about that typo.

So for the intersection, is when the two vectors intersect right? And this wants me to prove that their intersection is at the 0v. So can I do something like (x,y) - (a,b) = (0,0) <--(where x,y and a,b are two vectors, so the intersection is when they are at the same point, so their difference is 0?)
 
  • #6
1LastTry said:
Sorry about that typo.

So for the intersection, is when the two vectors intersect right? And this wants me to prove that their intersection is at the 0v. So can I do something like (x,y) - (a,b) = (0,0) <--(where x,y and a,b are two vectors, so the intersection is when they are at the same point, so their difference is 0?)

What are x,y, and and b, how do they relate to M and N, and why is the intersection of M and N at all related to subtraction?

A vector (x,y) is contained in the intersection of M and N if (x,y) is contained in M and (x,y) is contained in N. If (x,y) is contained in M, what do we know about x and y? If (x,y) is contained in N, what do we know about x and y?
 
  • #7
So M and N intersect when (x,y) is contained in both M and N...

if M = (x1, 0v2) and N = (0v1, x2) shouldn't the intersection calculated by when the difference of those is = 0?
So you can get the point where both M and N contains?
 
  • #8
M is not a vector, and neither is N so those equalities you wrote don't make sense.
 
  • #9
can we just assume there is a non-zero vector a1, a2 that's in both m and n for it to be in m, we know a1 = x 1 and a2 = 0 component vector thingie and since it's a non-zero vector, you know x1 does not equal the zero component vector now, apply a1,a2 to N since a1 is not a zero-component vector, a1 does not equal to the zero component, which is a requirement for the vector in N so a1 a2 is not in both sets which is a contradiction
so by proof of contradiction, there is no vector outside the zero vector in both m and n hence m intersection n is just the zero vector
 
  • #10
Yes, that sounds good to me. Technically you also should prove that the zero vector is contained in the intersection.
 

1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that also satisfies the properties of a vector space. This means that it must be closed under vector addition and scalar multiplication, and must contain the zero vector.

2. How do you determine if a subset is a subspace?

To determine if a subset is a subspace, you need to check if it satisfies the three properties of a vector space: closure under vector addition, closure under scalar multiplication, and containing the zero vector. If all three properties are satisfied, then the subset is a subspace.

3. Can a subspace be empty?

No, a subspace cannot be empty. It must contain at least the zero vector in order to satisfy the properties of a vector space.

4. What is the difference between a subspace and a span?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. On the other hand, a span is a set of all possible linear combinations of a given set of vectors. A subspace is a specific type of set, while a span is an operation on a set.

5. How can subspaces be used in applications?

Subspaces have many applications in various fields such as engineering, computer science, and physics. They can be used to solve systems of linear equations, find optimal solutions to problems, and analyze data in machine learning algorithms. In physics, subspaces are used to represent physical systems and their properties, such as energy levels in quantum mechanics.

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