Context
I am studying tensors. For me, tensors, their definitions, and their usage have always been unclear. They come up from time to time anecdotally. You might have seen something akin to, "This looks like a matrix, but its actually a tensor. Let's ignore that for now..." Yet, when I look deeper, like in [1] or [2], I am frankly befuddled by the volume of nomenclature to even state one (or the more than one possible) definitions of a tensor. Baring this in mind, rather than tackle the abstract, I'd like to lay down a few concrete anchors to help guide my learning process. In particular, considering that I am made to understand that a scalar is a tensor of type $(0,0)$ and that a Euclidean vector is a tensor of type $(1,0)$ [2]. I would like to do some examples to explicitly show that these are true.
I have made a preliminary search on this site. The closest questions that I have found are in [3] and [4]. In [3], Noah Schweber does a solid job of explaining what a scalar is in terms of fields and vector spaces. Yet, Schweber does not explain scalar in terms of a tensor of type $(0,0)$. In [3], J. W. Tanner mentions that "a second definition of a scalar is a quantity that transforms as a scalar (e.g. is unchanged) under a change of coordinates." This is certainly closer to the knowledge that I am seeking to understand. Nonetheless, Tanner states this anecdotally, and does not demonstrate that a scalar actually does what is proposed. In [4], RobbieFresh asks a similar question. I believe that, in response, spaceisdarkgreen expounding on the matter of how come there are no indices; and gives several trains of thought on that particular point. However, spaceisdarkgreen does not specifically adhere to a particular definition to explain how a tensor of type (0,0) transforms. In [4], Hans Lundmark offers a succint remark that again reinforces the point that there are no indeces in tensor of type (0,0), but Lundmark goes no further.
Questions
(1) How does one apply any of the definitions of the tensor found in [2], to show, step by step, that a scalar meets the definition of a tensor of type $(0,0)$?
(2) How does one apply any of the definitions of the tensor found in [2], to show, step by step, that a Euclidean vector meets the definition of a tensor of type $(1,0)$?
My First Attempt at a Solution
[Definition of Tensor from Ricci] A tensor of type $( p , q )$ is an assignment of a multidimensional array $$T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}]$$ to each basis $f = \left(\mathbf{e}_1 , \ldots, \mathbf{e}_1\right)$ of an $n$-dimensional vector space such that, if we apply the change of basis $$\mathbf{f}\mapsto \mathbf{f}\cdot R = \left( \mathbf{e}_i\, R^i_1, \dots, \mathbf{e}_i R^i_n \right)\,,$$ then the multidimensional array obeys the transformation law $$ T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}] R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q}\, . $$
Here goes...
Proposition: A scalar is a tensor of type (0,0)
Proof: For the case of a tensor of type (0,0) there are no indices at all. This is explained by Hans Lundmark in [4]. From the definition, a tensor of type $( 0 , 0 )$ is an assignment of a single-entry array $$T[\mathbf{f}].$$ to each basis $f = \left(\mathbf{e}_1 \right)$ of an $1$-dimensional vector space such that, if we apply the change of basis $$\mathbf{f}\mapsto \mathbf{f}\cdot R = \left( \mathbf{e}_1\, R^1_1 \right)\,,$$ then the single-entry array obeys the transformation law \begin{align} T [\mathbf{f} \cdot R] &= \left(R^{-1}\right)^1_1 \, T [\mathbf{f}]\, R^1_1 \\ &= T [\mathbf{f}] \, \left(R^{-1}\right)^1_1\, R^1_1 && \text{commutation} \\ &= T [\mathbf{f}] &&\text{inverse elements} \end{align} I observe that the tensor corresponding to the scalar has zero covariant indices and zero contravariant indices. I conclude that the tensor corresponding to the scalar is of type (0,0). Q.E.D.
Proposition: A Euclidean vector is a tensor of type (1,0)
Proof: ...
Bibliography
[1] CRC, 30th Edition, page 431.
[2] https://en.wikipedia.org/wiki/Tensor
