Row/Column space in relation to row operations

[Gridvvk]

I'm having trouble wrapping my head around what should be a trivial detail, but it is important, so hopefully someone else putting it in explicit words might help me understand it.

What I am having trouble grasping is why do row operations preserve linear dependence relations for the columns of a matrix but not the rows?

The context this comes up is in regards to the row & column space of a matrix. Given a matrix A, to find a basis for the column space we would just take the linearly independent columns of A. However, usually it's difficult to tell what columns are independent, so we find rref(A) and the pivot positions in rref(A) correspond directly to the pivot positions in A, this is true because row operations preserve linear dependence relations for the columns.

For the row space we would take the linearly independent rows of rref(A) this is because the row space of A is equivalent to rref(A); however, the dependence relations for the rows are not the same.

So if I can understand why the dependence relations are the same for columns but different for rows, it would really help me connect everything together.

 

[economicsnerd]

Let $A$ be an $n\times m$ matrix.
- Say you want to switch two rows of $A$ to create a matrix $B$. Is there an $n\times n$ matrix $E$ you can write down that will carry out the operation for you? i.e. Can you choose $E$ to ensure $EA=B$?
- Say you want to scale some row of $A$ by a nonzero constant to create a matrix $C$. Is there an $n\times n$ matrix $F$ you can write down that will carry out the operation for you? i.e. Can you choose $F$ to ensure $FA=C$?
- Say you want to add a multiple of one row of $A$ to another row, to create a matrix $D$. Is there an $n\times n$ matrix $G$ you can write down that will carry out the operation for you? i.e. Can you choose $G$ to ensure $GA=D$?

Now that you've figured out what $E,F,G$ all look like, do you notice any feature they all have? [Hint: Do they have any feature that will ensure, for instance, that $EA$ and $A$ have the same number of linearly independent rows?]

 

[Gridvvk]

Let $A$ be an $n\times m$ matrix.
- Say you want to switch two rows of $A$ to create a matrix $B$. Is there an $n\times n$ matrix $E$ you can write down that will carry out the operation for you? i.e. Can you choose $E$ to ensure $EA=B$?

Yes E would be the elementary matrix corresponding to the same operation on the identity matrix.

Say you want to scale some row of $A$ by a nonzero constant to create a matrix $C$. Is there an $n\times n$ matrix $F$ you can write down that will carry out the operation for you? i.e. Can you choose $F$ to ensure $FA=C$?

F would be the elementary corresponding to the same scaling on the identity matrix.

Say you want to add a multiple of one row of $A$ to another row, to create a matrix $D$. Is there an $n\times n$ matrix $G$ you can write down that will carry out the operation for you? i.e. Can you choose $G$ to ensure $GA=D$?

G would also be elementary matrix formed by doing the same operation on the n by n identity.

Now that you've figured out what $E,F,G$ all look like, do you notice any feature they all have? [Hint: Do they have any feature that will ensure, for instance, that $EA$ and $A$ have the same number of linearly independent rows?]

E,F, and G are elementary matrices. EA and A should have the same number of linearly independent rows, but why don't the same pivot positions for these rows in EA correspond to the pivot positions in A?

 

[Erland]

If you perform elementary column operations instead of row operations, then the linear relations between the rows are preserved, but not those between the colums. On the other hand, the column space is unchanged, but not the row space.

 

[Gridvvk]

If you perform elementary column operations instead of row operations, then the linear relations between the rows are preserved, but not those between the colums. On the other hand, the column space is unchanged, but not the row space.

Yes. That would be logically true, if you buy that row operations preserve linear relations between columns but not rows. You wouldn't even need to invent column operations you can instead claim the original proposition for the transpose of a matrix.

That still doesn't explain why it is true.