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According to the direct sum theorem of matroid, (1) the union of 2 matroids whose ground sets are disjoint and (2)whose independent set is defined as the union of the independent sets of the two respective matroids, is a matroid.

when we say two matroid in this context, do we refer to the same types of matroids?

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    Then the answer is no. The type of the matroids doesn't matter.2012-08-14

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Let's have a matroid built from a matrix and a matroid built from a graph and construct the new object and see if we encounter anything bothering us because of the difference of motivation of the two matroids or not.

Take $A$, a $2\times 2$ matrix as $\begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix}$ and Define the Graph $G$ be the complete graph on 3 vertices, meaning $K_3$ with edges called $a$, $b$ and $c$. Then if I call the matroids coming from them with $M_1$ and $M_2$ respectively, we have (index columns of $A$ with $1$ and $2$ from left to right);

$\begin{array}{l} E(M_1)=\{1,2\}\\ I(M_1)=\{\emptyset,\{1\}\}\\ E(M_2)=\{a,b,c\}\\ I(M_2)=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\} \end{array}$

Conditions of your sum theorem are hold.

$\begin{array}{l} E(M):=\{1,2,a,b,c\}\\ I(M)=\{\emptyset,\{1\},\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\} \end{array}$

As you can check readably, this is indeed a matroid.