# basis for matrix

• Jan 28th 2007, 08:39 AM
PvtBillPilgrim
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?
• Jan 28th 2007, 03:04 PM
PvtBillPilgrim
Could anyone just see if this makes sense?
• Jan 28th 2007, 05:04 PM
Soltras
Quote:

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.