Give a matrix, B, so that it's null space is a given set of vectors

[Easy_as_Pi]

Homework Statement

Give a matrix B so that the subspace W defined in part b (W = (1,1,0,-2),(1,-1,1,6),(0,1,1,4)) can be written as
W = N(B) where N(B) is the null space of B

Homework Equations

none that I know of, other than N(A) = {vectors x | Ax = 0}

The Attempt at a Solution

I have no clue where to begin. I did not know it was possible to reverse engineer 3 which represent a null space to find the original matrix. Any idea on how to start this problem? I'm utterly lost.

 

[fzero]

Can you find a vector that is orthogonal to all vectors in W? How could you relate this vector to a matrix?

 

[Easy_as_Pi]

If I let A be the matrix whose rows are given by the three vectors that span W, then find the null space of that matrix, the result would be all vectors x, such that Ax=0. Yet, this is just the null space of the subspace I've been given, not a null space of matrix such that the null space is the given subspace.

 

[Bacle2]

If you mean the subspace generated by the vectors in W*, then one way is:

* I'm assuming these vectors are L.I.

1)Extend the basis for W into a basis W^=W\/{w^} for the whole vector space.

2)Define a linear transformation T, with T(w)==0 for all vevtors in W

3) Define T(w^)=v , for v any vector in a basis for ℝ ⁴, and extend by linearity.

4)Find the matrix associated with T.

In 3), use the fact that a linear bijection is an isomorphisms; in this case, you have an

isomorphism between the 1-d vector subspace generated by w^, and the 1-d subspace

of ℝ⁴ generated by v. This means that the kernel of this last map is {0}.

 

[fzero]

Your comments about the matrix A are on point. In fact the null space of that matrix is important to solving your problem.

What can you say about the row vectors of the matrix B that you want to find? What space do they span?

 

[Bacle2]

Sorry if I was too abrupt; fzero , you seem much more patient (in a good way), than I am.

 

[Easy_as_Pi]

Hey sorry I haven't replied yet; I had work from 8am-7pm today. Bacle2, they are linearly independent, but I'm not quite sure what you mean by extending the subspace. That might not be something my class has covered yet (our book seems to go in a different order compared to note sets I have found online to supplement my studying.
Fzero, I hope you don't think I'm trying to get you to work out the problem for me; I'm just really confused by it. I know that the orthogonal complement to the row space of a matrix equals the null space of the same matrix, and I think I need to use that fact, but I'm not sure how.
I know that those three given vectors need to be orthogonal to every row in B, so the row vectors of B would be in R4. So those three vectors are the orthogonal complement to the row space of B, right?

 

[Bacle2]

I see. Maybe you're seeing the fundamental theorem of linear algebra:http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra

 

[fzero]

Hey sorry I haven't replied yet; I had work from 8am-7pm today. Bacle2, they are linearly independent, but I'm not quite sure what you mean by extending the subspace. That might not be something my class has covered yet (our book seems to go in a different order compared to note sets I have found online to supplement my studying.
Fzero, I hope you don't think I'm trying to get you to work out the problem for me; I'm just really confused by it.

No, I'm not worried. I'm just trying to give small hints without giving too much of the answer away,

I know that the orthogonal complement to the row space of a matrix equals the null space of the same matrix, and I think I need to use that fact, but I'm not sure how.
I know that those three given vectors need to be orthogonal to every row in B, so the row vectors of B would be in R4. So those three vectors are the orthogonal complement to the row space of B, right?

Yes, but you're just resaying that W = N(B). That's ok, but how can we use this information? Following along on Bacle2's last suggestion about the FTLA, can you determine the rank of B? How many linearly independent row vectors does B have? Can you determine the basis for the span of the row vectors of B?

Said another way, as you say, the elements of W need to be orthogonal to every row in B. Can you start trying to explicitly find vectors which are orthogonal to the elements of W?

You are probably going to be amazed at how straightforward this is once it hits you. That's not bad, since it's clear that you understand the things you need to. Solving problems like this is the best way to make that knowledge click.