Originally Posted by

**Deveno** personally, i'm a big fan of doing things in this order:

1). studying R^n and matrices for a little bit.

2). learning a little group theory.

3). learning linear algebra proper.

4). learning some more group theory, a little ring theory, and then tackling R-modules.

often the order of 2 and 3 is transposed, or 2 is omitted entirely. this often obscures some of the natural motivations for why we care about linear transformations, null spaces, subspaces and images spaces (span sets). abelian groups are very "nice" algebraic structures, and much of the regularity of linear algebra comes from this. in fact, i feel too much emphasis is placed on the underlying field in most people's first look at linear algebra, the only reason we want a field in the first place, is so we have enough algebraic rules for something like row-reduction to work. so even R has "too much structure", working over Q would be "good enough".