Hey,

The basic idea behind subspaces, to give you some intuition, is that anything you multiply or add together in the space stays in the space. That's pretty much all you need.

Let's get a general feeling for what these spaces would look like.

a) For this space, you know you want the things to look like (3, 3, 3, 3), (11441, 11441, 11441, 11441) -- anything whose components are equal. The important thing to imagine here is the geometry of what's going on. If you lower down to R3, you can imagine each vector having some direction and magnitude. To get to any point in R3 (that is, to have a basis for R3), you need three vectors that are linearly independent. Linear independence in this case means that all three vectors are going in completely different directions. So, for instance, if you used three pencils, and put two of your pencils flat on your desk and a third one sticking up, you would just have to make sure the two pencils lying flat weren't pointing in the same direction. If you put them perpendicular to one another, you have the familiar three axes (x, y, z).

So how do you determine the direction? Well, scaling certainly doesn't affect direction -- and this is true for any dimension. But you can see that (3, 3, 3, 3) = 3*(1, 1, 1, 1), for instance. This would lead us to believe our subspace is one-dimensional, and indeed it is. We have that all our vectors are just a multiple of (1, 1, 1, 1), so that any vector whose components are equal is a basis for this space.

Does this make sense?