1. ## polynomial subspaces

Which of the following subsets of $\displaystyle P_{2}$ are subspaces? The set of all polynomials of the form:
a. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{1}=0$ and $\displaystyle a_{0}=0$
b. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{1}=2a_{0}$
c. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{2}+a_{1}+a_{0}=2$

2. Originally Posted by antman
Which of the following subsets of $\displaystyle P_{2}$ are subspaces? The set of all polynomials of the form:
a. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{1}=0$ and $\displaystyle a_{0}=0$
b. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{1}=2a_{0}$
c. $\displaystyle a_{2}t^{2}+a_{1}t+a_{0}$, where $\displaystyle a_{2}+a_{1}+a_{0}=2$
To be a suspace we require for the set to be closed under addition and scalar multiplication.
Check if this is true for each of these cases.

3. So then a and b would be subspaces? C would not be because if it is multiplied by zero, the conditions a2+a1+a0=2 would not be satisfied. When b is multiplied by 0 to get 0+0+0, a_1=2_a0 because 0=2(0) and 0=0?

4. Originally Posted by antman
So then a and b would be subspaces? C would not be because if it is multiplied by zero, the conditions a2+a1+a0=2 would not be satisfied. When b is multiplied by 0 to get 0+0+0, a_1=2_a0 because 0=2(0) and 0=0?
Yes, , (a) and (b) are subspaces but (c) is not.