Finding Elements in Row Space of a Matrix: Solving for Belonging and Dependency

[EvLer]

In my homework problem, I am supposed to find out whether an element belongs to the rowspace of a matrix. So, what I did is to determine the (row)basis of the matrix, dimension of it being one row less of the rows of the original matrix. So, instead of the linearly-dependent row I put the element and if the system turns out to be inconsistent I assume that it does not belong to the row space.
Is it correct to assume that? If I use dependency equation for the rows and the new row element would that give the same result? When I used it I got there are no solutions at all, I am not sure what it means because for being lin. indep. there has to be one solution: 0.

Thanks in advance.

P.S. I decided to post the problem itself afterall:
2 1 3 1
1 1 3 0
0 1 2 1
3 3 8 2
and I need to determine whether X = [4, 1, 2, 5] and Y = [1, 2, 3, 4] belong to row space of the matrix.
The answer is X does, but Y does not.

 

[Galileo]

So you need to find a linear combination of the row vector which yields X or Y.
Putting the row vectors (or a basis for the row space) as columns in A, you're asked to solve:
Ax=X and Ax=Y, (lousy notation, but A is a matrix, x is the unknown vector and X,Y are given)

How would you normally solve such an equation?

 

[EvLer]

If I were asked to find out whether X belonged to the column space of a matrix, would I do the same thing, i.e. insert that column at the end (in augmented matrix) and see whether it is solvable?
Thanks a lot, my previous problem worked out!
And a follow-up question:
how are basis of rowspace and basis of column space related, besides the fact that dimensions are equal?

 

[Galileo]

If I were asked to find out whether X belonged to the column space of a matrix, would I do the same thing, i.e. insert that column at the end (in augmented matrix) and see whether it is solvable?

Yes. The way I look at Ax in this case is a linear combination of the column vectors of A. So Ax=b (for some vector b) has a solution if and only if b lies in the column space of A.

Thanks a lot, my previous problem worked out!
And a follow-up question:
how are basis of rowspace and basis of column space related, besides the fact that dimensions are equal?

Can't think of anything now.