Finding a basis for a subspace

In summary, the subspace given by the equations x1+x2+x3+x4=0 and x1-x2+2x3+x4=0 has a basis of (-1.5, 0.5, 1, 0) and (1, 0, 0, 1). To find an orthonormal basis, we can use the Gram-Schmidt process.
  • #1
JasonJo
429
2
Let U be a proper subspace of R^4 and let it be given by the equations:

1) x1+x2+x3+x4=0
2) x1-x2+2x3+x4=0

how do i find a basis for this subspace?

I got that (0,1,2,0) is one of the basis vectors since x2=2x3, therefore whatever we pick for x2, x3 will be twice that value.

i also got that x4=-x1-1.5x3, but does this require two more basis vectors or one?

ie, I'm asking, it seems that every x value can be determined once x1 and x2 are determined, therefore it should have 2 basis vectors, but i can't quite put it into that form
 
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  • #2
I think that you have 2 leading variables and 2 dependent variables (parameters), i.e. your solution subspace is in dimension 2, with 2 vectors generating it.
reduced row echelon form gives you
x1 + 1.5x3 + x4 = 0
x2 - 0.5x3 = 0

from the first equation you have
x1 = - 1.5x3 - x4
from the second one you have
x2 = 0.5x3

and x3, x4 are parameters, so you can let x3 = t and x4 = s

Now

(x1,x2,x3,x4) = (-1.5t+s, 0.5t, t, s) =
t(-1.5, 0.5, 1, 0) + s(1, 0, 0, 1)

so the vectors you need to find are:

(-1.5, 0.5, 1, 0) and (1, 0, 0, 1).

I hope I didn't mess up with the numbers...
 
  • #3
JasonJo said:
I got that (0,1,2,0) is one of the basis vectors since x2=2x3, therefore whatever we pick for x2, x3 will be twice that value.

It's 2*x2 = x3 actually
 
  • #4
ok i found that basis, now how do i find an orthonormal basis?
 

Related to Finding a basis for a subspace

1. What is a basis for a subspace?

A basis for a subspace is a set of linearly independent vectors that span the entire subspace. This means that any vector within the subspace can be written as a linear combination of the basis vectors.

2. How do you find a basis for a subspace?

To find a basis for a subspace, you can use the pivot columns from the reduced row-echelon form of the matrix representing the subspace. These pivot columns will form a set of linearly independent vectors that span the subspace.

3. Can there be more than one basis for a subspace?

Yes, there can be more than one basis for a subspace. However, all bases for a given subspace will have the same number of vectors, known as the dimension of the subspace.

4. Why is finding a basis for a subspace important?

Finding a basis for a subspace is important because it allows us to represent any vector within the subspace as a linear combination of a small set of vectors. This makes it easier to perform calculations and solve problems involving vectors in that subspace.

5. Can a subspace have an infinite number of vectors in its basis?

Yes, a subspace can have an infinite number of vectors in its basis. This occurs when the subspace is a continuous space, such as the set of all polynomials of a certain degree. In this case, the dimension of the subspace is also infinite.

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