# Dimension of polynomial degree less than equal to 2 and polynomial which vanish at 0.

Let $$V$$ is vector space of polynomial of degree $$\leq$$ 3,$$W_1$$ be subspace of polynomial of degree $$\leq$$ ,2$$W_2$$ is subspace of polynomial vanishing at zero. then, \

1.Span of $$W_1,W_2$$ is V

2.$$W_1$$ is isomorphic to $$W_2$$

3.$$Dim(W_1 \cap W_2)$$ is 2

4.$$Dim(W_1) < Dim(W_2)$$

element of V will be of type $$a + bx + cx^2 +dx^3$$

$$W_1$$ ={ $$a + bx + cx^2$$ s.t$$a,b,c \in \mathbb {R}$$}

$$W_2$$ ={ $$bx + cx^2 +dx^3$$ s.t$$b,c,d \in \mathbb {R}$$}

here dimension of $$W_1$$ is three and that $$W_2$$ is also three. so 2nd option should be true.

4th option is wrong. 3rd is correct if we take their intersection basis will contain two elements namely{ $${x,x^2}$$}.

An element of $$V$$ can be written as $$a+bx+cx^2+dx^3$$. Notice that $$a+bx+cx^2$$ belongs to $$W_1$$ since it is of degree $$2$$, and $$dx^3$$ belongs to $$W_2$$ since it vanishes at $$0$$. So I think option 1 is also correct, since you can write anything in $$V$$ as a linear combination/sum of elements from $$W_1$$ and $$W_2$$.