# Are these statements regarding isomorphism of a linear mapping True or False?

I have to prove or disprove following statements, but I'm not completely sure if I am going in the right direction. If someone could please tell me if they are true or false, it would help me a lot. The statements are:

$1)$ Let $L : V \to W$ be a linear mapping and let $\{\vec{v}^{\,1}, \ldots,\vec{v}^{\,n}$} be a basis for $V$. If $\{L(\vec{v}^{\,1}), \ldots,L(\vec{v}^{\,n})\}$ spans $W$ then $L$ is isomorphic to $W$.

I think this statement is false because although $L$ is onto, but it is not one-to-one, so $L$ is not invertible.

$2)$ Let $L : V \to W$ and $M : W \to U$ be linear mappings. If $\dim V = \dim U$ and $M \circ L$ is onto, then $V$ and $W$ are isomorphic.

I think this statement is similar to the first one, except that it talks about composition of two linear mappings. But I'm not too sure.

Thanks for any hints.

1) Remember that $L$ is linear. Since $L$ is linear and $V$ and $W$ are both finite dimensional, then $L$ can be represented by a matrix. What does this matrix look like? What properties does it have?

• I think L should be a square matrix with no. of rows = n, since it is the number of vectors in the basis of V. May 23, 2018 at 16:31
• @kronos And what properties does that matrix have? How would you get its columns? May 23, 2018 at 16:45
• We should get a B-matrix of L, by calculating L(v_i) and writing it as a linear combination of vectors in basis of B, doing this for all the vectors in the basis of B. I can't think of a property which would help in this question. May 23, 2018 at 17:58
• @kronos Is B one-one? onto? May 23, 2018 at 19:01

part 1 - FALSE :

since {$L(v_1),...,L(v_n)$} spans $W$, we are not sure if it is a basis for $W$ or not. The given statement can only be true if {$L(v_1),...,L(v_n)$} is a basis for $W$.

part 2 - TRUE :

since the dimensions for $V$ and $U$ are equal so they are isomorphic to each other. But not all linear mappings $L:V -> U$, from V to U are isomorphic.