Can anyone please clarify why P_2 must have a degree ≤ 2 for it to be a subspace of P, the vector space of all polynomials?
Alright, I know that W is a subspace of V, a vector space, if W is also a vector space with respect to the operations in V. I also know that W is a subspace of V if and only if the following conditions hold:
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 $\displaystyle P_2$ (polynomials of degree 2 or less) within $\displaystyle P_3$ (polynomials of degree 3 or less). We can try to show that $\displaystyle P_2$ is a subspace of $\displaystyle P_3$. We can use the conditions you provided. To rephrase, we must check:
- Adding any two vectors in $\displaystyle P_2$ will again be a vector in $\displaystyle P_2$ (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 $\displaystyle P_2$ by a number will again be a vector in $\displaystyle P_2$ (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?
No, they do not. Hence, $\displaystyle P_{3}$ is not a subspace of $\displaystyle P_{2}$. Thank you for explaining this. I finally understand.
One thing I don't understand, however, is the fact that although $\displaystyle P_{3}$ is not a subspace of $\displaystyle P_{2}$, it is a subset of $\displaystyle P_{2}$, no? If that is the case, how does one determine if a polynomial $\displaystyle P_{n+1}$ is a subset of $\displaystyle P_{n}$?
Wait a second. We aren't showing that $\displaystyle P_2$ is a subspace of $\displaystyle P_3$? (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.)
That was my mistake, apologies. Ok, so $\displaystyle P_{2}$ is a subspace of $\displaystyle P_{3}$ if it holds for the two operations of vector addition and scalar multiplication, but what about $\displaystyle P_{3}$ being a subspace of $\displaystyle P_{2}$? Why would $\displaystyle P_{3}$ be considered a subset of $\displaystyle P_{2}$ but not a vector space? Is it due to the fact that adding two polynomials of degree two or less does not yield $\displaystyle P_{3}$?