how to find the basis of a subspace

Question

How to observe a basis for a subspace? (linear albebra / matrix theory)?

Ok here's the problem.

Detect a basis for the subspace Southward of R^4 consisting of all vectors of the form (a+b, a-b+2c, b, c)^T, where a,b,c are existent numbers. What is the dimension of Southward?

Im utterly confused with bases and dimensions...

Answers (ii)

Randy P

• Oct 25 'xx

If a subspace has dimension n, then a basis for that subspace is any set of n linearly independent vectors.

Based on the fact that at that place are three gratuitous parameters in the subspace S (a, b and c) which tin be varied independently, I'd guess that the dimension of S is 3. At whatsoever rate, it is no larger than 3.

Let me see if I can give a solid reason for that intuition... OK, I claim that the 1st, 2nd and 4th elements of (a+b, a-b+2c, b, c) can exist annihilation at all (or any other 3 elements, but information technology's easiest to see with those). That is, at that place is a solution to a + b = ten, b = z, c = westward for any choice of x, z, west.

But one time you that, you've fixed the 2nd element a - b + 2c. Then you don't have four independent components. One of them is dependent on the other iii.

As for choosing a basis for this three-dimensional subspace, just choose three independent iii-vectors as x, z and westward, find the corresponding values of a, b and c, and use that to specify the y value. Your choices need merely exist linearly independent, they don't have to be orthogonal. For instance (1, 1, 0), (1, 2, 0) and (0, 0, i) would do.

Edit: I've been away from this stuff besides long. The answer higher up is much more elegant and meaty. Practice you encounter why that works? Because you can limited any vector in S every bit a sum of a, b and c times those vectors, and because those 3 vectors are linearly contained, and then

the dimension of the subspace is 3 and
those vectors are a basis

They aren't the merely possible basis. That'south important to remember.

Accepted Answer

Your footing would be

(1,1,0,0), (1,-ane,1,0) and (0,2,0,1). Since there are 3 footing vectors, information technology'south a three-dimensional infinite.

Vectors of this form should be spanned past the footing, and then assume that a, b, and c are the coefficients in the linear combination, then yous have a(1,ane,0,0) + b(ane,-1,1,0) + c(0,2,1,0), which gives you the form you are looking for.

Edit: The answer immediately beneath mine is very adept, too. The point he brings upwardly about the basis not being unique is important.