Deleted User's Maths Thread

[Deleted User]

Is the proof for whether a set is linearly independent and whether it spans a subspace the same? If you put the set into a matrix as columns, reduce it and show that its rank is equal to the number of columns, then you've shown that it is both linearly independent and that it spans right? Or am I missing something?

[kamil9876]

I don't quite understand your question. If you are given a set of vectors of some vector space then it could be possible that they span and aren't linearly independent. It is also possible that they are linearly independent but do not span . Hence of course these are two SEPERATE tasks and so you should expect the proofs to be different.

However there is one special case, (and perhaps this is what you are referring to?) if you have vectors in (i.e number of vectors = dimension), then they are linearly indepenedent if and only if they span the whole of , so in this case it does suffice to just row reduce and see if the matrix has rank .

[Deleted User]

I don't quite understand your question. If you are given a set of vectors of some vector space then it could be possible that they span and aren't linearly independent. It is also possible that they are linearly independent but do not span . Hence of course these are two SEPERATE tasks and so you should expect the proofs to be different.

However there is one special case, (and perhaps this is what you are referring to?) if you have vectors in (i.e number of vectors = dimension), then they are linearly indepenedent if and only if they span the whole of , so in this case it does suffice to just row reduce and see if the matrix has rank .

For example, for this set: {(-1,1,2),(3,3,1),(1,2,2)} (R^3). If I find the rank of the row echelon form of this set, I get rank =3. As the rank is equal to the number of columns, the set is linearly independent and it also spans R^3 by definition right?

[kamil9876]

the set is linearly independent and it also spans R^3 by definition right?

It's a fact that three linearly independent vectors in span , I mentioned that in my previous post. You should look up the definition of "span", not sure if you really know what it is.

How about {(-1,1,2),(3,3,1)} ? Is it linearly independent? does this span ?

[Deleted User]

But for my question above, if I show that the set is linearly independent, do I have to do anything extra to show that it also spans?

[Deleted User]

And also how do I write this set as a matrix? {, , , } (M_2,2.)

[Deleted User]

What is the definition of a set that spans a subspace and a set that is linearly independent?

Using the augmented matrix form of a set, how do I prove that it spans a subspace and how do I show that it's linearly independent?

Thanks

[Alwin]

What is the definition of a set that spans a subspace and a set that is linearly independent?

I think you mean the basis of a subset yeah

Using the augmented matrix form of a set, how do I prove that it spans a subspace and how do I show that it's linearly independent?

You reduce the matrix into row reduced echelon form. For any pivot columns (ie all zeros except one 1 in one row) the vectors in the original columns are linearly independent. if the rank of the matrix is equal to the number of columns then the set of vectors is the basis of the subset and linearly independent.

It makes more sense with an example, so if you have a specific question I can run through it with you

[Deleted User]

How to I prove that a matrix, M, is idempotent using 2 different methods?

One is to show that MM=M

What's another method?

[kamil9876]

Too vague, do you have a specific problem/matrix in mind?

[Deleted User]

Ok I have to show that this is idempotent using 2 diff methods
4/9 4/9 2/9
4/9 4/9 2/9
2/9 2/9 1/9

So let M be the matrix above. One method is to multiply the matrix by itself and show hthat the answer is itself? So MM=M.

Do you know another method?

[Deleted User]

I need to prove that a matrix M is idempotent using 2 different methods.

One method I've used is to show that M*M=M.

What's another method I could use?

[kamil9876]

Is it still that particular problem?

Ok I have to show that this is idempotent using 2 diff methods
4/9 4/9 2/9
4/9 4/9 2/9
2/9 2/9 1/9

So let M be the matrix above. One method is to multiply the matrix by itself and show hthat the answer is itself? So MM=M.

Do you know another method?

[pi]

Split topic.

[Deleted User]

Is it still that particular problem?

Yes.