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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{ ${x,x^2}$}.

what about first?

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1 Answer 1

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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 $W_2$.

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  • $\begingroup$ yes i think you are correct. Actually my booklet has given answer as only option C, which i think is wrong , A and B are also correct. $\endgroup$ Commented Mar 22, 2022 at 7:28

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