Expressing matricies as vectors

[SprucerMoose]

G'day,

I'm doing some questions on whether or not a given set spans a given vector space and was wondering what the best way to write out a matrix is.

For example, if I am wanting to show that a bunch of M_2X2 elements spans M_2X2, can I express each matrix as a coordinate vector with respect to a basis say B={E^i,j|i=1,2;j=1,2}.

I have been writing my sets along the lines of...

S_B = {(1,2,3,4), (2,3,4,5),...}

to make it easier to write, but I'm not sure if this is the correct notation. Is is acceptable rewrite the whole set with respect to a basis like this? What is the best way to resolve matrices into a simpler form for calculation in such a scenario?

Also, in the general case, if I represent a bunch of vectors with respect to some other basis, if these vectors are linearly independent and/or span a subspace in the new basis, does that imply that, with respect to the original basis, the vectors span the same subspace and are also independent?

Thanks

 

[SprucerMoose]

Anybody?

 

[micromass]

G'day,

I'm doing some questions on whether or not a given set spans a given vector space and was wondering what the best way to write out a matrix is.

For example, if I am wanting to show that a bunch of M_2X2 elements spans M_2X2, can I express each matrix as a coordinate vector with respect to a basis say B={E^i,j|i=1,2;j=1,2}.

I have been writing my sets along the lines of...

S_B = {(1,2,3,4), (2,3,4,5),...}

to make it easier to write, but I'm not sure if this is the correct notation. Is is acceptable rewrite the whole set with respect to a basis like this? What is the best way to resolve matrices into a simpler form for calculation in such a scenario?

Let's say that there is certainly nothing wrong with the notation, it is mathematically valid. However, if I were you, I would indicate that these vectors are with respect to the basis B. So I would write it as

[tex]S_8=\{(1,2,3,4)_B,(2,3,4,5)_B,...\}[/tex]

just to indicate that we're not working with 4-tuples here, but rather with coordinates. Other then that, there is nothing mathematically wrong with this notation.

But, mathematically wrong is not the only thing you should be aware of. When handing is an assignment or something else, I would advice to write the matrices in their usual form and not in coordinate form. The reason is that matrices in their usual form are conceptually easier to understand, and we know how to handle them. Writing them as coordinates may confuse the reader...

Also, in the general case, if I represent a bunch of vectors with respect to some other basis, if these vectors are linearly independent and/or span a subspace in the new basis, does that imply that, with respect to the original basis, the vectors span the same subspace and are also independent?

Yes, spanning and linear dependence are notions which do not depend on the basis. So a linear independent set in one basis will be linear independent in every bas!s!

 

[SprucerMoose]

Thanks very much.

If I was to attempt to work with a group of nxn matrices, wouldn't it be easier to express them in coordinate form to check for independence, i.e. straight into an augmented matrix as columns and obtain RREF?

Also, can you work with a bunch of coordinate vectors for matrices as a column, row or null space, or is this an operation reserved for n-tuples? Is the column space, etc, of the M^n,n vector space defined. By the way how is an n-tuple defined? Is it just one of Rⁿ or Cⁿ.

Thanks again.

 

[micromass]

If I was to attempt to work with a group of nxn matrices, wouldn't it be easier to express them in coordinate form to check for independence, i.e. straight into an augmented matrix as columns and obtain RREF?

Yes, in this case it much easier to work in coordinate form!

Also, can you work with a bunch of coordinate vectors for matrices as a column, row or null space, or is this an operation reserved for n-tuples? Is the column space, etc, of the M^n,n vector space defined.

I have a hard to figuring out what you mean with this. Can you perhaps give an example of what you want to do?

By the way how is an n-tuple defined? Is it just one of Rⁿ or Cⁿ.

For your purposes, it is enough to know that an n-tuple is indeed just an element of [tex]\mathbb{R}^n[/tex] or [tex]\mathbb{C}^n[/tex]. It's just a collection of n elements where the order matters.

There is an entire set theoretical definition of n-tuples, but I fear that this will only confuse you. The important thing isn't how an n-tuple is defined, but what it's properties are...

 

[SprucerMoose]

I have a hard to figuring out what you mean with this. Can you perhaps give an example of what you want to do?

To be honest, I'm not even sure what I mean. If I obtain a bunch of vectors from M^n,n with respect to the bases decribed earlier, they will now be coordinate vectors, can I take these coordinate vectors and form a column space of coordinate vectors, or is this not "allowed" because they are not n-tuples?

 

[micromass]

Well, it's probably allowed to do such things. But I kind of doubt that such things will be useful. In any case, I've never seen people do this. But I don't think it would be wrong of you if you would treat coordinate vectors as n-tuples...

 

[SprucerMoose]

No worries.

Thanks again for you time and excellent responses.