Finding a basis and dimension of a subspace

[black_89gt]

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 :(

 

[VeeEight]

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 a_i'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)

 

[black_89gt]

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.

 

[VeeEight]

I'm not sure what you mean when you say 'the variable c3 and c4 depend on each other' but see this link:
http://en.wikipedia.org/wiki/Linear_independence#Formal_definition

You are going to want to solve the system of equations and see if the coefficients are all zero, as in the definition in the link.

 

[black_89gt]

When I do this, a1v1 + ... + a5, v5 = 0, i end up getting -a1=4a4; a3=a4=a5=0;

 

[HallsofIvy]

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.