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Let's say I have a finite one-dimensional vector space $V$ with basis $\{\alpha\}$. Then the product vector space $V^k$ has basis $\{e_1, \dots, e_k\}$ where $e_i = (0, \dots, \alpha, \dots, 0)$ where $\alpha$ is in the $i$-th position in the $k$-tuple.

Let's say I have a function $f : V^k \to V$ defined in some way $f(x_1, \dots, x_k) = \dots$

Now the $x_1, \dots x_k$ are variables above, and are just placeholders for values really. Also technically $f(x_1, \dots, x_k)$ is shorthand for $f((x_1, \dots, x_k))$. That being said is it correct to view $(x_1, \dots, x_k)$ as $x_1\cdot e_1 + x_2\cdot e_2 + \dots + x_k\cdot e_k$ or as $(x_1 \alpha, \dots, x_k\alpha)$?

Is it thus correct to think of $f(x_1, \dots, x_k)$ as $f(x_1\cdot e_1 + x_2\cdot e_2 + \dots + x_k\cdot e_k)$ or as $f(x_1\alpha, \dots, x_k\alpha)$?

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    $\begingroup$ I cannot make sense out of the equation $e_i=(0,0,...,\alpha,...,0)$ where $\{\alpha\}$ is a basis. $\endgroup$ – Kavi Rama Murthy Jan 15 at 5:48
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    $\begingroup$ @KaviRamaMurthy I think the OP means that $\{\alpha\}$ is a basis of $V$, so a basis $\{e_i\}_{i=1}^{k}$ of $V^k$ can be constructed by placing $\alpha$ in the $i^\text{th}$ entry, and having the zero element of $V$ as the others. $\endgroup$ – user445909 Jan 15 at 9:16
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Suppose $V$ is a vector space over a field $\mathbb{F}$.

Since the domain of $f$ is $V^k$, each entry of the $k$-tuple $(x_1,\dots,x_k)$ seen in $f(x_1,\dots,x_k)$ is a vector in $V$. Otherwise, $f(x_1,\dots, x_k)$ is not defined. Thus, for every $i \in \{1,\dots , k \}$, the product $x_i \alpha$ is between two members of $V$, which is not a standard operation on a vector space. Since this operation is undefined, we cannot make sense of $f(x_1 \alpha, \dots, x_k \alpha)$ as the situation stands.

However, if we define the coordinate map $C: \mathbb{F}^k \to V^k$, where for every $(x_1,\dots,x_n) \in \mathbb{F}^k,$ we define $C(x_1,\dots,x_n) := (x_1 \alpha, \dots, x_n \alpha)$, then we could write \begin{align} (f\circ C)(x_1,\dots,x_n) &= f(x_1 \alpha,\dots,x_n \alpha) \\ &= f(x_1 e_1 + \dots + x_n e_n), \end{align} under the usual definition of entry-wise vector addition and scalar multiplication in $V^k$.

In short, in order to make sense of $f(x_1,\dots,x_k)$ as you expressed it, each $x_i$ should be thought of as a scalar, not a vector. But the domain of $f$ is $V^k$, so we cannot do this unless we define a coordinate map.

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