- #1
boneill3
- 127
- 0
Homework Statement
Let {e1,e2,e3} be a basis for the vgector space V over the field F.
Put f1 = -e1, f2 = e1+e2 and f3 =e1 + e3
Prove that {f1,f2,f3} is also a basis for V
Homework Equations
The Attempt at a Solution
I made e1,e2,e3 be the unit bases.
1
0
e1= 0
0
0
e2= 1
0
0
e3= 1
which makes {f1,f2,f3}
-1
0
f1= 0
1
1
f2= 0
1
0
f3= 1
so {f1,f2,f3} =
-1 1 1 0
a1 * 0 + a2 * 1 + a3 * 0 = 0
0 0 1 0
What i did was set this matrix equal to zero and solve.
The solution was only the trivial solution.
As this proves that {f1,f2,f3} is linear independent is that enough to show that it also is a basis ?
regards