Exactness of sequence of vector spaces with tensor product

Question

Let $k$ be a field and $0\rightarrow V _n \xrightarrow{f_n} V_ {n − 1} \xrightarrow{f_{n−1}} ··· \xrightarrow{f_3} V_ 2 \xrightarrow{f_2} V_ 1 \xrightarrow{f_2} V_ 0 \rightarrow 0$ an exact sequence of finite-dimensional $k$ -vector spaces.

Let $V$ be a finite-dimensional k -vector space. Is the sequence

$0\rightarrow{V_n\otimes V}\xrightarrow{f_n\otimes 1_V}V_{n-1}\otimes V\xrightarrow{f_{n-1}\otimes 1_V}\cdots \xrightarrow{f_3\otimes 1_V}{V_2\otimes V}\xrightarrow{f_2\otimes 1V}V_1\otimes V\xrightarrow{f_0\otimes 1_V}V_0\otimes V\rightarrow 0$

necessarily exact?

Answer 1

Yes, every vector space is a 'flat module'.