# If $\varphi|_W$ and $\bar{\phi}$ are nonsingular prove that $\varphi$ is nonsingular.

Here is the question I am trying to answer the second and the third part of it:

If $$W$$ is a subspace of the vector space $$V$$ stable under the linear transformation $$\varphi$$(i.e., $$\varphi(W) \subseteq W$$), show that $$\varphi$$ induces linear transformations $$\varphi |_{W}$$ on $$W$$ and $$\bar{\varphi}$$ on the quotient vector space $$V/W.$$ If $$\varphi|_W$$ and $$\bar{\phi}$$ are nonsingular prove that $$\varphi$$ is nonsingular. Prove the converse holds if $$V$$ has finite dimension and give a counterexample with $$V$$ infinite dimensional.

I know that a linear transformation $$f$$ is nonsingular iff ker f = 0, but still I do not know how to prove the second and the third part of the above question. Any hint will be greatly appreciated.

Edit: Here is my trial depending on the given comments below:

Since we know that a linear transformation $$f$$ is nonsingular iff $$\ker f = 0.$$ and since we are given that ker $$\bar{\varphi} = 0$$ and ker $$\varphi|_W =0$$ and since my definition for $$\bar{\varphi}(x)$$ is $$\varphi (x) + W$$ then $$x \in ker \bar{\varphi}$$ implies that $$\bar{\varphi}(x) = W$$ which means that $$\varphi(x) \in W.$$ but then how can I use the assumption that ker $$\varphi|_W =0$$? I know that from this assumption, I can conclude that $$\varphi(w) = 0$$ iff $$w=0$$ but then how can I complete the proof till the end? Could someone help me please?

• Here is a hint for the second part. Suppose that $x \in \ker\phi$. What can you say about $\overline{\phi}(x+W)$, and can you use that to show $x \in W$? Dec 3, 2023 at 22:07
• Related fact: if $V$ is finite-dimensional, $\det(\varphi)=\det(\varphi|_W)\det(\overline{\varphi})$. Dec 3, 2023 at 22:19
• @blargoner why is that correct? How this will help me in the proof? Dec 4, 2023 at 0:27
• @ChrisEagle I have edited my question, do you have a justification for the question I asked in the edit? Dec 4, 2023 at 0:43
• @blargoner I have an edit for my question that contains a question, do you have an answer for the question in the edit? Dec 4, 2023 at 0:45

1. If $$\varphi|_W$$ and $$\overline{\varphi}$$ are both nonsingular, suppose that $$\varphi(x)=0$$. What can you say about $$\overline{\varphi}(\overline{x})=\overline{\varphi(x)}$$, where here $$\overline{z}=z+W\in V/W$$. What does this tell you about $$\overline{x}$$? Why does that tell you that $$x\in W$$? Finally, why does that tell you that $$x=0$$?
2. If $$V$$ is finite-dimensional and $$\varphi$$ is nonsingular, why is it immediate that $$\varphi|_W$$ is also nonsingular? Why does this tell you that $$\varphi(W)=W$$? Now suppose $$\overline{\varphi}(\overline{x})=0$$ where $$x\in V$$. Why does this tell you that $$\varphi(x)\in W$$? Why does this tell you that $$x\in W$$? Finally, why does that tell you that $$\overline{x}=0$$?
3. Consider the space $$V$$ of infinite sequences $$(x_1,x_2,\ldots)$$ and $$\varphi:V\to V$$ given by $$\varphi(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$$.