1. ## basis for matrix

I have this matrix made of two spanning sets:
1 1 3 2 0 8
5 4 14 3 1 9
-2 0 -4 -2 0 -8
3 3 9 0 1 3

In reduced row echelon form the matrix is:
1 0 2 0 1/6 0
0 1 1 0 1/6 0
0 0 0 1 -1/6 0
0 0 0 0 0 1

W1 is the span of the first three columns of the original matrix.
W2 is the span of the last three columns of the original matrix.

I need to find a basis for:
W1
W2
W1+W2
the intersection of W1 and W2

I think I'm wrong but I said that:
a basis for W1= {Column 1, Column 2}
a basis for W2={Column 4, Column 5}
a basis for W1+W2={Column 1, Column 2, Column 4, Column 6}
this means that there would be no basis for the intersection of W1 and W2

doesn't make sense
could someone check this?

2. Could anyone just see if this makes sense?

3. Originally Posted by PvtBillPilgrim
I have this matrix made of two spanning sets:
1 1 3 2 0 8
5 4 14 3 1 9
-2 0 -4 -2 0 -8
3 3 9 0 1 3

In reduced row echelon form the matrix is:
1 0 2 0 1/6 0
0 1 1 0 1/6 0
0 0 0 1 -1/6 0
0 0 0 0 0 1

W1 is the span of the first three columns of the original matrix.
W2 is the span of the last three columns of the original matrix.

I need to find a basis for:
W1
W2
W1+W2
the intersection of W1 and W2

I think I'm wrong but I said that:
a basis for W1= {Column 1, Column 2}
a basis for W2={Column 4, Column 5}
a basis for W1+W2={Column 1, Column 2, Column 4, Column 6}
this means that there would be no basis for the intersection of W1 and W2

doesn't make sense
could someone check this?
I don't have time to work it all out, but looks like you're right on W1 and W1+W2, except for W2 you need all three columns because those three are all linearly independent.

From my initial glance, it looks like a basis for the intersection is the single vector <1,1,0,0> (not an actual column from ths matrix). This is because W1 has vectors looking like <a,b,0,0> and W2 has vectors looking like <c,c,d,e>.

Sorry I don't have time to explain in more detail. I hope this helps.