# Finding a basis and dimension of a subspace

## 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)?

## 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 :(

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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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.

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

HallsofIvy