Linear Algebra - subset spanning R^4

  • #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.
 

Answers and Replies

  • #2
34,667
6,379

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.
 

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