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**1. Homework Statement**

Which of the following sets are subspaces of

**##R[x]?##**

##W_1 = {f \in \mathbf R[x] : f(0) = 0}##

##W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}##

##W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}##

##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n a_ix^2i}##

**2. Homework Equations**

The criteria for a given set being a subspace are that it be closed under addition and scalar multiplication, and also that the 0 vector belongs to the set.

**3. The Attempt at a Solution**

##W_1## is a subspace:

##f(0)+g(0) = (f+g)(0) \Rightarrow 0 + 0 = 0 \in W_1##

##\lambda \cdot f(0) = (\lambda \cdot f(0)) = \lambda \cdot 0 = 0 \in W_1##

##0v = 0 \in W_1##

##W_2## is not a subspace:

##2f(0) = f(1) \neq (2 \cdot f(0)) \Rightarrow## The condition of closure under scalar multiplication is not met.

I also suspect the 0 vector does not belong to this set, but I am unsure of how to show this.

The notation and executing the notation correctly is my biggest problem here, however, I appreciate all conceptual insight. I have thus far not been able to correctly describe whether or not ##W_3## and ##W_4## are subspaces. My suspicion is that ##W_3## is not a subspace because ##f(0) = f(1-0) = f(1) \neq f(0)## but I am uncertain of how to improve this train of thought. Similarly, I believe ##W_4## meets all the criteria to be a subspace of ##\mathbf R[x]## but I do not know how to correctly use notation to describe how it meets those criteria.

Additionally, I would like clarification about what exactly ##\mathbf R[x]## refers to.

Thank you very much.