# Is a Basis unique?

• May 13th 2011, 09:08 AM
sj247
Is a Basis unique?
When trying to find a basis, I got (-3, -11, 1, 0 ) & (3, 1, 0, 10)

When i saw an example using a different technique to me, (3, 1, 0, 10) & (3, 0, 1, 11) was the basis?

Is that possible, generally and in this example? Thanks
• May 13th 2011, 09:15 AM
spiral
A basis is not unique.
As long as your basis spans the vector space, and the elements in the basis are linearly independent, then your basis will work; in general, we usually try to find the smallest or simplest basis that will work.
• May 14th 2011, 02:24 AM
Deveno
Quote:

Originally Posted by spiral
A basis is not unique.
As long as your basis spans the vector space, and the elements in the basis are linearly independent, then your basis will work; in general, we usually try to find the smallest or simplest basis that will work.

all bases have the same "size" (literally if V is finite-dimensional).

in this case, however, one of the bases must be incorrect: (3,0,1,11) is not in span{(-3,-11,1,0),(3,1,0,10)}.

if it were, we would have (3,0,1,11) = a(-3,-11,1,0) + b(3,1,0,10), giving us:

-3a + 3b =3
-11a + b = 0
a = 1
10b = 11 --> b = 11/10. but -11 + 11/10 is not 0, so we cannot find any such a and b.
• May 14th 2011, 06:40 AM
HallsofIvy
"Ackbeet: Deveno beat me"

Do you have that on a rubber stamp?(Rofl)
• May 14th 2011, 08:06 AM
spiral
Quote:

-3a + 3b =3
-11a + b = 0
a = 1
10b = 11 --> b = 11/10. but -11 + 11/10 is not 0, so we cannot find any such a and b.
I think your "a" is wrong.
I got a = 1/10, b = 11/10 which seems to work out.
• May 14th 2011, 02:09 PM
Deveno
the 3rd coordinate of (3,0,1,11) is 1. the 3rd coordinate of a(-3,-11,1,0) + b(3,1,0,10) = (-3a+3b,-11a+b,a,10b) is a.

hence, if the two are to be equal, a MUST EQUAL 1. the 4 equations

-3a+3b = 3
-11a+b = 0
a=1
10b=11

must all have a SIMULTANEOUS solution for (3,1,0,11) to be in the span.