Finding a basis and dimension of a subspace

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  • #1

Homework Statement



Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]}

Let W=spanS. Find a basis for W. What is dim(W)?


Homework Equations



The Attempt at a Solution



i know that a basis is composed of linearly independent sets. This particular problem's basis cannot be greater than 4 since the last row of all of the column vectors is 0. i have looked all throughout my book and online through this site and i cannot figure out how to even start this problem. Please help me :(
 

Answers and Replies

  • #2
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You want to find a set that spans W and consists of linearly independent vectors. Are the vectors {v1, ..., v5} linearly independent? Use the method of setting a1v1 + .... + a5, v5 = 0, where the ai's are scalars, to check this.

i know that a basis is composed of linearly independent sets.

A basis is a set composed of linearly independent vectors (and it must also span the space)
 
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  • #3
no, they arent because the variable c3 and c4 depend on each other in row 2 of the matrix after setting c1v1+c2v2+c3v3+c4v4+c5v5=0.
 
  • #5
When I do this, a1v1 + .... + a5, v5 = 0, i end up getting -a1=4a4; a3=a4=a5=0;
 
  • #6
HallsofIvy
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The set of all [a, b, c, d] is four dimensional so 5 vectors can't be independent. It should be easy to see that v2= 4v1 so you can just drop v2. v5 is independent of all the others because it has a "1" in the third place while all others have "0".

Finally, v4= [2, -1, 0, 0]= 2[1, 0, 0, 0]- 1[0, 1, 0, 0]= 2v1- v3 so you can drop v4. v1 and v3 are obviously independent.
 

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