General questions about intersection of subspaces

[talolard]

Hey Guys.
I have some questions about vector spaces, I would really apreciate if someone could read this and let me know if I understand things or not, and if not let me know where I have it wrong.

I am having a lot of trouble UNDERSTANDING how to find the intersection of two vector spaced and how to find the space of solutions of a system of equations.

Given two spaces and there bases as vectors I think the way to find the intersection is to write a matrix with each vector as a column. This is because I am looking for a linear combination of these vectors that equals 0. Since if it equals 0 then there is an equivalence between the vectors.
Having taken this matrix and brining it to reduced row echelon form, what do I do from here? More importantly, do I understand this coorectly?

With a system of solutions: Given a number of equations I would put each equation in the matrix as a row and bring it to reuced row echelon form. This is because in this case I am searching for the dependencies between variables in the equations by combining them. So if I have a single 1 in each row and everything else zero than that variable is equal to the coresponding entry in the solutions column.
If there is a row with two variables then the opening variable (the first entry in that row with a 1) is tied to the other variable.
I just need to know if I understand this or not as when I try to solve problems I am unsure of myself because I am not sure if i know what I am doing or not.
Thanks
Tal

 

[Mark44]

Hey Guys.
I have some questions about vector spaces, I would really apreciate if someone could read this and let me know if I understand things or not, and if not let me know where I have it wrong.

I am having a lot of trouble UNDERSTANDING how to find the intersection of two vector spaced and how to find the space of solutions of a system of equations.

I think you really mean two subspaces of a vector space.

Given two spaces and there bases as vectors I think the way to find the intersection is to write a matrix with each vector as a column. This is because I am looking for a linear combination of these vectors that equals 0. Since if it equals 0 then there is an equivalence between the vectors.
Having taken this matrix and brining it to reduced row echelon form, what do I do from here? More importantly, do I understand this coorectly?

Make that "Given two subspaces of a vector space ..."

I don't think your plan will work. Let me make things a little less abstract. Let's say that one of the subspaces is U and a basis for U is {u₁, u₂}. Let's say that another subspace is W and a basis for it is {w₁, w₂}. You create a matrix whose columns are u₁, u₂, w₁, and w₂ and row reduce it. If you find that the only solution for the coefficients of u₁, u₂, w₁, and w₂ is the zero vector, all you have established is that the four vectors are linearly independent. This does not help you in determining the intersection of the two subspaces.

One approach that might be helpful is to determine whether there is some linear combination of the basis vectors in one subspace that produces one of the basis vectors in the other subspace. You could do this for each vector in the other subspace. If you get a solution, then the given vector is in the intersection of the two subspaces. If you don't get a solution, then the given vector is not in the intersection.

With a system of solutions: Given a number of equations I would put each equation in the matrix as a row and bring it to reuced row echelon form. This is because in this case I am searching for the dependencies between variables in the equations by combining them. So if I have a single 1 in each row and everything else zero than that variable is equal to the coresponding entry in the solutions column.
If there is a row with two variables then the opening variable (the first entry in that row with a 1) is tied to the other variable.
I just need to know if I understand this or not as when I try to solve problems I am unsure of myself because I am not sure if i know what I am doing or not.
Thanks
Tal

 

[talolard]

Thanks,
That's exactly what I needed to hear.
Tal