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Let $V=\{a_0+a_1X+\cdots+a_3X^3: a_j \in \mathbb{Q}\}$, a vector space over $\mathbb{Q}$ under the usual polynomial addition and scalar multiplication. Let $W=\operatorname{span}(\{X+1,2X-1,X^2+X\})$. Find the spanning set for V/W. Prove that your answer is correct.

I am sure that this is a simple question. The one thing that is confusing me is how to deal with quotient space. Any help is great.

Thanks in advance.

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1 Answer 1

up vote 1 down vote accepted

Let's see:

$$a(x+1)+b(2x-1)+c(x^2+x)=0\Longleftrightarrow cx^2+(a+2b+c)x+a-b=0\Longleftrightarrow$$ $$ c=0\,\,,\,\,(a=b)\wedge (a=-2b)\Longrightarrow b=a=-2b\Longrightarrow b=a=0$$

Thus, we've shown the set $\,\{x+1\,,\,2x-1\,,\,x^2+x\}\,$ is linearly independent and since $\,\dim V=4\,$ , we get $\,\dim V/W=1\,$ as $\,\dim W= 3\,$ .

Since clearly $\,x^3\notin W\,$ , we can take $\,V/W=\operatorname{Span}\{x^3+W\}\,$

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