Hi, i am struggling with the idea of basis and dim.
Find a basis for the subspace S of R3 spanned by
{ v1 = (1,2,2), v2 = (3,2,1), v3 = (11,10,7), v4 = (7,6,4) }.
What is dim s ?
A basis is a spanning set that is also independent. You are given that this subspace is spanned by these four vectors so a basis would be a subset of that. Start with any one of them, say, v1.
v2 is not a multiple of v1 so {v1, v2} are independent.
Now look at v3. Is v3 independent of v1 and v2? That is, can v3 be written as a linear combination of v1 and v2? That is the same as asking "can we have av1+ bv2+ cv3= 0 where a, b, c are not all 0?".
If yes, add v3. If no, ask the same about v4.
Since $\displaystyle R^3$ itself has dimension 3, a basis cannot contain more than 3 vectors.