I'm trying to evaluate $\mathbf{M}(\vec{e}_x)$, where
$$\mathbf{M}=\tilde{\omega}^x\otimes\vec{e}_x +\tilde{\omega}^x\otimes\vec{e}_y+\tilde{\omega}^y\otimes\vec{e}_y.\tag{1}$$
Here's my first question. Am I correct in thinking that this is a $\begin{pmatrix}1\\1\end{pmatrix}$ tensor because the dimensionality cancels on the first and third terms on the right-handside?
Assuming that that is the case, then $\mathbf{M}$ can be expressed as
$$ \mathbf{M}=M_i^{\;j}\,\tilde{\omega}^i\otimes\vec{e}_j, $$ so that $$ \begin{align*} \mathbf{M}(\vec{e}_x)&=\mathbf{M}(\vec{e}_x\,;\ \,)\\ &=M_i^{\;j}\,\tilde{\omega}^i(\vec{e}_x)\vec{e}_j\\ &=M_i^{\;j}\,\delta^i_{\;x}\,\vec{e}_j\\ &=M_x^{\;j}\,\vec{e}_j\\ &=M_x^{\;x}\,\vec{e}_x+M_x^{\;y}\,\vec{e}_y. \end{align*} $$ Now, is it correct to think of $M_i^{\;j}$ as a $(2\times2)$ matrix? Assuming so, inspection of $(1)$ gives $$ \mathbf{M}(\vec{e}_x\,;\ \,)=\vec{e}_x+\vec{e}_y. $$ Is this correct? I'm relatively new to tensors so any help will be appreciated!