Let W be the subspace of R5 spanned by 1 2 0 0 0 0 2 3 0 0 0

Let W be the subspace of R^5 spanned by [1 2 0 0 0], [0 2 3 0 0], [0 0 0 4 5], [1 0 0 0 5]. Then the dimension of W is

Solution

First we check for linear independence of given vectors,

a[1 2 0 0 0]+b[0 2 3 0 0]+c[0 0 3 4 0]+d[0 0 0 4 5]+e[1 0 0 0 5]=0

This gives us

a+e=0 ie a=-e

2a+b=0 ie b=-2a=2e

3b+3c=0 ie c=-b=-2e

4c+4d=0 ie d=-c=2e

5d+5e=0 ie d=-e=2e

Hence e=0 and hence a=b=c=d=0

Hence these are 5 linearly independent vectors

So dimension of W is

E.5