Isomorphism(one-to-one and onto linear mappings)

[helenli89]

I don't really understand what is isomorphism and how to use it. I thought isomorphism as a linear mapping that is one-to-one and on-to.
I have tried to prove the following:
"Let L: U->V and M: V->W be linear mappings. Prove that if L and M are 1:1, then M(L(x)) or M ? L is 1:1"
I proved that C1U1+...+CnUn = 0 has only the trival solution (lamar of 1:1) therefore L(C1U1+...+CnUn) = C1LU1+... +CnLU1=C1V1+...CnVn = 0 has only the trival solustion since L and M are 1:1. Therefore M ? L is 1:1 as well.
I don't know if this make sense or not and if it's too short of a proof.

I have also come across another question:
"Prove that if U and V are finite-dimensional vectors spaces over R, then U and V are isomorphic if and only if (iff) they are the same dimension."
I know that since an Isomorphic mapping has to be 1:1 and onto, which mean that it's bijection so every image has exactly one pre-image or in another word every equation have exactly one solution. Also a theorem states that a linear mapping L: U->V has an iverse linear mapping L^(-1): V->U iff L is 1:1 and onto. Therefore a if U and V are finite dimensional vector spaces then they are isomorphic iff they have the same dimension.
But I don't think this is a good proof, it's more like an explaination. I don't know how to write a proper proof for this.

Helen

 

[qpmathelp]

"Let L: U->V and M: V->W be linear mappings. Prove that if L and M are 1:1, then M(L(x)) or M ? L is 1:1"


suppose u1 and u2 belong to U such that
M ? L (u1) = M ? L (u2)

implies M [ L(u1) ] = M [ L(u2) ]

implies L(u1) = L(u2) since M is one to one

implies u1 = u2 since L is one to one

implies M ? L is one to one

 

[helenli89]

"Let L: U->V and M: V->W be linear mappings. Prove that if L and M are 1:1, then M(L(x)) or M ? L is 1:1"

suppose u1 and u2 belong to U such that
M ? L (u1) = M ? L (u2)

implies M [ L(u1) ] = M [ L(u2) ]

implies L(u1) = L(u2) since M is one to one

implies u1 = u2 since L is one to one

implies M ? L is one to one

Thanks but it's too late I already know how to do it and handed in my assignment and now I already got my marked assignment back.
I don't see your way of proving this question. At the first line "M ? L (u1) = M ? L (u2)" already implies u1=u2 since the linear mapping is same at both side of the question and the only difference is the vector that's been mapped and the equal sign surggests that not only the linear mapping is the same but also the vectors. So i don't think your prove is strong enough; but again I'm not the marker.
Here is the solution:

 

[helenli89]

"Prove that if U and V are finite-dimensional vectors spaces over R, then U and V are isomorphic if and only if (iff) they are the same dimension."

Answer to this question is as follows: