# Writing a tensor with respect to a particular basis

When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:

$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu \otimes f^\rho$$

where $\{e_\mu\}$ is a basis for the vector space, and $\{f^\mu\}$ is its dual basis.

I know how to expand the tensor when applied to some generic vectors and covectors. So in the example above:

$$T(\alpha,\beta,X) = T(\alpha_\mu f^\mu,\beta_\nu f^\nu, X^\rho e_\rho) = \alpha_\mu \beta_\nu X^\rho T(f^\mu,f^\nu,e_\rho) = \alpha_\mu \beta_\nu X^\rho\, T^{\mu\nu}_{\rho}$$

How would I proceed from here? Can I just say $\alpha_\mu = \alpha_\nu f^\nu(e_\mu) = \alpha (e_\mu)$, and similarly for the others? Then I would get:

$$T(\alpha,\beta,X) = T^{\mu\nu}_{\rho}\, \alpha(e_\mu)\, \beta(e_\nu)\, f^\rho(X)$$

I am not sure how to generalise from here. Apologies if this is really obvious.

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