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Prove that $\eta: V \to V/W$ defined as $\eta(x)=x+W$ is a surjective linear transformation, and $\ker(\eta)=W$.

My proof: Let $y \in V/W$. Then let $x=y+W$. So $\eta(x)=\eta(y+W)=(y+W)+W=y+W$

I don't think that last step is right at all. I'm not too sure how to find the kernel either.

Thanks in advance.

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3 Answers 3

up vote 2 down vote accepted

The elements of $V/W$ are the sets of the form $x+W$ for $x\in V$, so if $y\in V/W$, then $y$ already is $x+W$ for some $x\in V$. That means that $y=\eta(x)$, so we’ve just proved that $\eta$ is surjective. You still have to prove that $\eta$ is linear and find $\ker\eta$.

To show that $\eta$ is linear, you have to show that for every $x,y\in V$ and scalars $\alpha,\beta$, $$\eta(\alpha x+\beta y)=\alpha\eta(x)+\beta\eta(y)\;.$$ To do that, use the definition of $\eta$ to write down what $\eta(\alpha x+\beta y)$ and $\alpha\eta(x)+\beta\eta(y)$ are, and see if you can see (and explain!) why they must be equal.

The kernel of $\eta$ is the set of vectors $x\in V$ such that $\eta(x)$ is the zero vector in $V/W$. What is that zero vector? Once you figure out what it is, it’s not to hard to figure out which elements $x+W$ of $V/W$ are equal to it.

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Hint: What is the zero vector of $V/W$?

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Is it just $0+W=W$? –  tk2 Sep 24 '12 at 22:56
    
@tkrm Mhm...knowing that, what must the kernel be? –  AsinglePANCAKE Sep 24 '12 at 22:57
1  
The kernal must be W then. –  tk2 Sep 24 '12 at 22:59

Your proof needs slight modification. Let $y \in V/ W$. By definition of what it means to be an element of $V / W$ we may choose $x \in V$ such that $y = x + W$. Then $\eta(x) = x+W = y$ which shows $\eta$ is surjective.

Then we need to show that $\ker(\eta) = W$. Firstly, we must consider what the zero vector in $V / W$ is. It is the equivalence class of $0\in V$. Hence, it is $W$. Let any $x \in W$ be given, then we certainly have $\eta(x) = x+W=W$ so $\ker(\eta) \supseteq W$. Alternatively, suppose $x \in V$ but $x \notin W$, then $\eta(x) = x+W \neq W$, so $x \notin \ker(\eta)$. It follows that $\ker(\eta) = W$.

Note: In the previous proof we used the fact that $x +W = W \iff x \in W$ since $W$ is a subspace.

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