there's nothing special about 2, we can define a similar space of polynomials of degree ≤ n, for any natural number n.
1. if u and v are any vectors in W, then the vector addition of u and v is in W.
2. if c is any real number and u is any vector in W, then the scalar multiplication of c with u is in W.
W is a subspace of V if and only if it is closed under these two operations.
I also understand that every vector space has at least two subspaces, itself and the subspace 0 consisting only of the zero vector.
I understand all of this, just when they went into this other rule is where the confusion begins.
What "other rule" is confusing you?
To go back to the polynomial spaces example, let's try to consider the space (polynomials of degree 2 or less) within (polynomials of degree 3 or less). We can try to show that is a subspace of . We can use the conditions you provided. To rephrase, we must check:
- Adding any two vectors in will again be a vector in (i.e. if I add two polynomials of degree two or less, the result is again a polynomial of degree two or less)
- Multiplying any vector in by a number will again be a vector in (i.e. if I multiply a polynomial of degree two or less by a number, the result is again a polynomial of degree two or less)
Do these two conditions hold?
One thing I don't understand, however, is the fact that although is not a subspace of , it is a subset of , no? If that is the case, how does one determine if a polynomial is a subset of ?
Wait a second. We aren't showing that is a subspace of ? (You have written it in the reverse.) If the conditions fail, could you please give a counterexample? (That is, if you say that adding two polynomials of degree two or less does not necessarily give another polynomial of degree two or less, give an example.)