How can I prove that W contains all elementary matrices?

In summary: What I first did was to try to find a matrix that when multiplied by the matrix w (which is in W, I thought) would give a matrix with all 0's except for some 1 in the i,j position. And I was trying to get a contradiction from there. But with w not necessarily invertible, this is not the way to go. So I like your proof better. I'm sorry I called it "unnecessarily complicated." It is not.
  • #1
rudo
2
0
Please help me proove the following:

Let V be a vector space over all n-by-n square matrices. Let W be a non-trivial subspace of V satisfying the following condition: if A is an element of W and B is an element of V then AB, BA are both elements of W.

Proove that W = V.

And here is what I am thinking about it...

1. If W contains the identity matrix then this equivalence is quite obvious. Whatever matrix B from V you give me, I multiply it by the identity matrix I and the B I = B is also element of W.

2. When W contains a regular matrix it is quite similar - because then W must contain the identity matrix. Let A be a regular matrix from W, then if you give me an inverse matrix, when I multiply them then what I will get is the identity matrix and due to the aforesaid condition the identity matrix is an element of W.

But somehow I do not know how to continue... Please help
 
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  • #2
I don't know if this will be helpful, but try to prove that W is a subset of V (which is trivial), and V is a subset of W in order to prove the equality.

Edit: corrected.
 
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  • #3
is this some deliberate attempt to misspell the word 'prove'?

W is non trivial. Pick w=/=0 in W. Now just show you can generate all of V. Hint: pick some obvious basis of V, show every element of this basis can be obtained by multiplying w by some appropriate choices.
 
  • #4
Matt, your methods seems unnecessarily complicated. The method rudo outlined, together with radou's hint works nicely. At first I puzzled over this because I misread the problem- I thought that W and V were subspaces of some other space! Since W is a subspace of V and V is the space of all 4 by 4 matrices, If w is any invertible matrix in W, then w-1 is contained in V and so ww-1= I is contained in W. from that, if v is any matrix in V then Iv= v is in W.

Hmm, suppose there are no invertible matrices in W? Is it possible a set of non-invertible matrices to be a subspace?
 
  • #5
HallsofIvy said:
Is it possible a set of non-invertible matrices to be a subspace?

Of course. For example, span{M}, where M is invertible, is an example.

We must show that a subspace consisting only of non-invertible matrices (which must be of the form span{M1, M2,...Mk}, where M1, M2,...Mk are each non-invertible) cannot satisfy the property "if A is an element of W and B is an element of V then AB, BA are both elements of W." Unfortunately a product of a non-invertible matrix with any matrix is still non-invertible, making the contradiction non-immediate. However, a sum of non-invertible matrices needs not be non-invertible.

The case of a 1-dimensional such subspace is ruled out since wv cannot be a multiple of w for all v. I believe a subspace, of dimension greater than 1, consisting only of non-invertible matrices cannot exist. Right?

I'm beginning to think Matt Grime's method is perhaps easier. Take w and multiply it by elementary matrices from V to get the matrix with 1 in the ijth entry and 0's everywhere else. Obtaining a basis for V that also exists in W. Hence W=V. The property that "AB, BA are BOTH elements of W." is needed to get the appropriate row and column operations to achieve the matrix transformations.
 
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  • #6
Unnecessarily complicated? No, sorry, I have to disagree there. My method takes about 3 lines if you try hard to make it long winded.

1. W contains a non-zero element
2. W contains an elementary matrix
3. W contains all elementary matrices.
 
  • #7
HallsofIvy said:
Hmm, suppose there are no invertible matrices in W? Is it possible a set of non-invertible matrices to be a subspace?

Yes. Trivially, and very important ones they are too: the set of strictly upper triangular matrices.

Of course, none that are two sided ideals.

If I wanted to be fancy I would have asked you to prove that the only automorphisms of M_n are inner, and that M_n is a simple algebra.
 
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  • #8
Thanks all of you for your help...
 
  • #9
matt grime said:
Unnecessarily complicated? No, sorry, I have to disagree there. My method takes about 3 lines if you try hard to make it long winded.

1. W contains a non-zero element
2. W contains an elementary matrix
3. W contains all elementary matrices.

Having realized that the subspace does not necessarily contain an invertible matrix, I now agree with you!
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements known as vectors, which can be added together and multiplied by scalars (usually real numbers). It follows a set of axioms that define its properties, such as closure under addition and scalar multiplication, and the existence of a zero vector and additive inverses.

2. What is a vector space proof?

A vector space proof is a mathematical demonstration that uses logical reasoning and the axioms of vector spaces to show that a given set is indeed a vector space. It typically involves showing that the set satisfies all the necessary properties and follows the rules of vector spaces.

3. Why is vector space proof important?

Vector space proof is important because it allows us to formally establish that a given set is a vector space, which in turn allows us to apply the properties and theorems of vector spaces to solve problems and make predictions in various fields of mathematics and science.

4. How do I approach a vector space proof?

When approaching a vector space proof, it is important to carefully read and understand the axioms and definitions of vector spaces. Then, start by assuming that the set in question is a vector space and try to prove that it satisfies all the necessary properties. If you encounter any difficulties, try to find a counterexample to the property in question, which would disprove the assumption that the set is a vector space.

5. Are there any tips for successfully completing a vector space proof?

Some tips for completing a vector space proof include carefully organizing your steps and using clear and concise language, as well as double-checking your work for any errors. Additionally, it can be helpful to break down the proof into smaller, more manageable steps and to seek help or clarification if you are stuck on a particular part.

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