Linear Algebra - subset spanning R^4

In summary: But since the vectors in S are already linearly independent, no need to find a subset of S that is a basis for the subspace spanned by S.
  • #1

Homework Statement



Let S={(1,2,3,4),(-1,2,0,1),(1,0,1,1),(-2,2,1,0)}. Determine whether or not S is linearly independent. If not, write one of the vectors in S as a linear combination of the others. Find a subset of S which is a basis of the subspace of R4 spanned by S.


The Attempt at a Solution


I put the 4 vectors into matrix form, and reduced it to get:
[1 2 3 4]
[0 2 2 3]
[0 0 -1 -1]
[0 0 0 -2]
So they are linearly independant. Since S has a rank of 4, it spans R4. So any basis of the subspace of R4 should be spanned by S.
My subset of S which is a basis of the subspace of R4 and spanned by S is:
{(1,0,0,0),(0,1,0,0),{0,0,1,0),{0,0,0,1)}

Is this correct? I'm not quite sure what they mean by a subset of S.
 
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  • #2
PirateFan308 said:

Homework Statement



Let S={(1,2,3,4),(-1,2,0,1),(1,0,1,1),(-2,2,1,0)}. Determine whether or not S is linearly independent. If not, write one of the vectors in S as a linear combination of the others. Find a subset of S which is a basis of the subspace of R4 spanned by S.


The Attempt at a Solution


I put the 4 vectors into matrix form, and reduced it to get:
[1 2 3 4]
[0 2 2 3]
[0 0 -1 -1]
[0 0 0 -2]
So they are linearly independant. Since S has a rank of 4, it spans R4. So any basis of the subspace of R4 should be spanned by S.
My subset of S which is a basis of the subspace of R4 and spanned by S is:
{(1,0,0,0),(0,1,0,0),{0,0,1,0),{0,0,0,1)}

Is this correct? I'm not quite sure what they mean by a subset of S.
Yes, the vectors in S are linearly independent, so they span R4.

Their question was hypothetical, in case the vectors in S were linearly dependent. If that had been true they wanted you to find the largest set of linearly independent vectors in S, which would have been a basis for some subspace of R4.
 

1. What is a subset spanning R^4?

A subset spanning R^4 is a set of vectors that, when combined, can form any vector in the vector space R^4. This means that the subset contains enough vectors to span the entire 4-dimensional space.

2. How do you determine if a subset spans R^4?

To determine if a subset spans R^4, you can use the span test. This involves taking a linear combination of the vectors in the subset and checking if it can create any vector in R^4. If it can, then the subset spans R^4.

3. Can a subset spanning R^4 have fewer than 4 vectors?

No, a subset spanning R^4 must have at least 4 vectors because R^4 is a 4-dimensional vector space. Any fewer than 4 vectors would not be enough to span the entire space.

4. Is a subset spanning R^4 unique?

No, a subset spanning R^4 is not unique. There can be multiple subsets of vectors that can span R^4, as long as they contain at least 4 linearly independent vectors.

5. How is linear independence related to a subset spanning R^4?

A subset spanning R^4 must contain linearly independent vectors. This means that none of the vectors in the subset can be written as a linear combination of the others. If a subset is not linearly independent, it cannot span R^4.

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