# Finding a Subset of S that is a Basis for <S>

• jeffreylze
In summary, the conversation discusses finding a subset of a given set that serves as a basis for a particular subspace. The process involves forming a matrix and row reducing it to row echelon form, with the corresponding columns on the leading entry being the desired basis. The symbol <S> represents the subspace spanned by the vectors in the original set, with the question simply asking for a basis for this subspace. The resulting basis, {(1,-1,2,1),(0,1,1,-2)}, is a subset of the original set.
jeffreylze

## Homework Statement

After sorting out basis, now I encountered a new term, Subset. And I get all confused again, clearly I'm not good at these, but I am learning.

Let S = {(1,-1,2,1) , (0,1,1,-2) , (1,-3,0,5)}
Find a subset of S that is a basis for <S>

## The Attempt at a Solution

I formed a matrix and row reduced it to row echelon form. Then the corresponding columns on the leading entry gives the basis which is {(1,-1,2,1) ,(0,1,1,-2)} . But now, to answer the question, how to find a subset ? and also, what does <S> mean ? Is that a symbol for a basis?

S is the original set of three vectors. <S> means the subspace spanned by the vectors in S. When you did your matrix exercise you found a basis for the subspace consisting of {(1,-1,2,1) ,(0,1,1,-2)}. That IS a subset of S. You are all done. The question was just to find a basis for the subspaces consisting of vectors from S. That's all.

## 1. What is a basis for a subset?

A basis for a subset is a set of vectors that can be used to form any other vector in the subset through linear combinations.

## 2. How do I know if a subset has a basis?

A subset has a basis if it is linearly independent, meaning that no vector in the subset can be formed by a linear combination of the other vectors in the subset.

## 3. How can I find a basis for a given subset?

To find a basis for a given subset, you can use the Gram-Schmidt process to orthogonalize the vectors in the subset. The resulting orthogonal vectors will form a basis for the subset.

## 4. Can a subset have more than one basis?

Yes, a subset can have multiple bases. This is because there are often many different sets of vectors that can form a basis for a given subset.

## 5. Is a basis unique for a subset?

No, a basis for a subset is not necessarily unique. It is possible for a subset to have multiple bases, and different scientists may use different methods to find a basis for the same subset.

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