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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.

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2 Answers 2

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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?

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  • $\begingroup$ 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. $\endgroup$
    – kronos
    May 23, 2018 at 16:31
  • $\begingroup$ @kronos And what properties does that matrix have? How would you get its columns? $\endgroup$
    – NicNic8
    May 23, 2018 at 16:45
  • $\begingroup$ 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. $\endgroup$
    – kronos
    May 23, 2018 at 17:58
  • $\begingroup$ @kronos Is B one-one? onto? $\endgroup$
    – NicNic8
    May 23, 2018 at 19:01
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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.

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