I have a question that is really confusing for me. In Serge Lang, the elements of $K^n$ where $K$ is a field are called vectors. The thing is that the vector $V=(v_1,...,v_n)$ belongs indeed to $K^n$ but the vector $P=\begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}^T=V$. The definition of $K^n$ is the set of $n-tuples$ of $K$, so $P$ and $V$ belongs to $K^n$ and they are well known as vectors in $K^n$. My problem is to prove that this set is a vector space. Please help me. How can you make addition between $V$ and $P$ if they have another way to write them regardless they are called vectors on the same vector space $K^n$.
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$\begingroup$ $P$ is not equal to $V$. It belongs to the dual of $K^n$, which is set theoretically distinct. $\endgroup$Dan Rust– Dan Rust2025-11-27 11:23:23 +00:00Commented Nov 27 at 11:23
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$\begingroup$ It is really confusing since the author says that they are elements of $K^n$ called as vectors. $\endgroup$William Avila Aguilar– William Avila Aguilar2025-11-27 11:28:02 +00:00Commented Nov 27 at 11:28
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$\begingroup$ To which of Lang's books do you refer? $\endgroup$Paul Frost– Paul Frost2025-11-27 11:34:33 +00:00Commented Nov 27 at 11:34
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$\begingroup$ Serge Lang, Linear Algebra Third edition but I think as $V$ and $P$ belong in fact to $K^n$ which means that the belong to the same vector space. $\endgroup$William Avila Aguilar– William Avila Aguilar2025-11-27 11:42:01 +00:00Commented Nov 27 at 11:42
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2$\begingroup$ You should either think of $K^n$ as being the set of row vectors or the set of column vectors, not both at the same time. $\endgroup$Dan Rust– Dan Rust2025-11-27 11:46:04 +00:00Commented Nov 27 at 11:46
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