# Proof of The Third Isomorphism Theorem

Mathematics Asked by Abhijeet Vats on August 12, 2020

Here’s what I’m trying to prove right now:

Let $$V$$ be a vector space over $$mathbb{F}$$. Let $$M$$ be a linear subspace of $$V$$ and $$N$$ be a linear subspace of $$M$$. Prove that the mapping $$x+N mapsto x+M$$ between the quotient spaces $$V/N to V/M$$ is linear with kernel $$M/N$$. Then, deduce that:

$$frac{V/N}{M/N} cong V/M$$

Proof Attempt:

We have to show that the given relation $$T: V/N to V/M$$ is a linear mapping. We define:

$$forall x in V: T(x+N) = x+M$$

This is totally-defined. To show well-definedness, let $$x+N = y+N$$ where $$x,y in V$$. Then, $$x-y in N$$. So, $$x-y in M$$. Hence:

$$x + M = y+M$$

$$iff T(x+N) = T(y+N)$$

To prove that it is linear, we need to show additivity and homogeneity.

1. Proof of Additivity

Let $$u,v in V/N$$. Then, $$u = x+N$$ and $$v = y+N$$ for some $$x,y in V$$. Then:

$$T(u+v) = T((x+N)+(y+N)) = T((x+y)+N) = (x+y)+M = (x+M)+(y+M) = T(u) + T(v)$$

1. Proof of Homogeneity

Let $$alpha in mathbb{F}$$ and $$u in V/N$$. Then, $$u = x+N$$ for some $$x in V$$. So:

$$T(alpha u) = T(alpha(x+N)) = T(alpha x +N) = alpha x + M = alpha (x+M) = alpha T(u)$$

This proves homogeneity. Hence, $$T$$ is a linear map. To show that the kernel of $$T$$ is $$M/N$$, we have:

$$T(x+N) = x+M = theta_V+M$$

$$iff x in M$$

$$iff x+N in M/N$$

$$iff ker(T) = M/N$$

Now, we notice that $$T$$ is surjective. By the first isomorphism theorem, it follows that:

$$frac{V/N}{M/N} cong V/M$$

That proves the desired result.

Does the proof above work? If it doesn’t, why? How can I fix it?

Your approach is absolutely right!

I would change the part concerning $$ker T$$, writing: begin{align} x + N in ker T & iff T(x+N) = 0 in V/M \ & iff x+M = 0 in V/M \ & iff x in M \ & iff x + N in M/N end{align} which means $$ker T = M/N$$.

Correct answer by Rodrigo Dias on August 12, 2020

## Related Questions

### How do you take the derivative $frac{d}{dx} int_a^x f(x,t) dt$?

1  Asked on December 1, 2021 by klein4

### Uniqueness of measures related to the Stieltjes transforms

1  Asked on December 1, 2021

### Does there exist a function which is real-valued, non-negative and bandlimited?

1  Asked on December 1, 2021 by muzi

### Suppose $A , B , C$ are arbitrary sets and we know that $( A times B ) cap ( C times D ) = emptyset$ What conclusion can we draw?

3  Asked on November 29, 2021 by anonymous-molecule

### Can a nonsingular matrix be column-permuted so that the diagonal blocks are nonsingular?

1  Asked on November 29, 2021 by syeh_106

### Does ${f(x)=ln(e^{x^2})}$ reduce to ${x^2ln(e)}$ or ${2xln(e)}$?

3  Asked on November 29, 2021 by evo

### Proving that $(0,1)$ is uncountable

2  Asked on November 29, 2021 by henry-brown

### Understanding a statement about composite linear maps

1  Asked on November 29, 2021

### Assert the range of a binomial coefficient divided by power of a number

3  Asked on November 29, 2021 by vib_29

### What is the Fourier transform of $|x|$?

3  Asked on November 29, 2021

### Exists $t^*in mathbb{R}$ such that $y(t^*)=-1$?.

2  Asked on November 29, 2021 by user514695

### Proving $logleft(frac{4^n}{sqrt{2n+1}{2nchoose n+m}}right)geq frac{m^2}{n}$

2  Asked on November 29, 2021 by zaragosa

### Linearized system for $begin{cases} frac{d}{dt} x_1 = -x_1 + x_2 \ frac{d}{dt} x_2 = x_1 – x_2^3 end{cases}$ is not resting at rest point?

1  Asked on November 29, 2021 by user3137490

### Is it possible to construct a continuous and bijective map from $mathbb{R}^n$ to $[0,1]$?

3  Asked on November 29, 2021 by kaaatata

### If $lim_{ntoinfty}|a_{n+1}/a_n|=L$, then $lim_{ntoinfty}|a_n|^{1/n}=L$

2  Asked on November 29, 2021 by diiiiiklllllll

### Right adjoint to the forgetful functor $text{Ob}$

1  Asked on November 29, 2021 by alf262

### Always factorise polynomials

1  Asked on November 29, 2021 by beblunt

### Formulas for the Spinor Representation Product Decompositions $2^{[frac{N-1}{2}]} otimes 2^{[frac{N-1}{2}]}=?$ and …

0  Asked on November 29, 2021

### Connected and Hausdorff topological space whose topology is stable under countable intersection,

1  Asked on November 29, 2021

### Evaluating an improper integral – issues taking the cubic root of a negative number

1  Asked on November 29, 2021

### Ask a Question

Get help from others!