Basis and Dimension

[TheWrathOfMath]

I was asked to find the basis and dimension of the linear subspace of the solution to the equation Ax=0.

I changed the matrix into row echelon form and found that:

(x1, x2, x3, x4) = (8x2, x2, 0, 0)

How do I proceed from here?

Do I write it as x2(8,1)+1(0,0)?
Do I write it as x2(8,1,0,0)?

 

[blamocur]

I like you second suggestion better than the first, if only because the first refers to vectors in [imath]\mathbb R^2[/imath], but you need [imath]\mathbb R^4[/imath].

So what is the dimension of the subspace?

And the basis?

 

[TheWrathOfMath]

I like you second suggestion better than the first, if only because the first refers to vectors in [imath]\mathbb R^2[/imath], but you need [imath]\mathbb R^4[/imath].

So what is the dimension of the subspace?

And the basis?

Then I suppose that the basis is (8,1,0,0) and the dimension is 4? Is that right?

 

[blamocur]

Then I suppose that the basis is (8,1,0,0) and the dimension is 4? Is that right?

No, not right yet. I agree that (8,1,0,0) can be used as a basis vector, but how many (linearly independent) basis vectors are there in your subspace?

 

[TheWrathOfMath]

No, not right yet. I agree that (8,1,0,0) can be used as a basis vector, but how many (linearly independent) basis vectors are there in your subspace?

Please excuse me for the idiotic error. The dimension is one since there is only one vector.