I saw the following description in a book of Algebraic Geometry, as example of affine variety.
Let $l_1,…,l_k $ be independent linear forms in $X_1,…,X_n$. Let $a_1,…,a_k \in \mathbb{C} $.Then $X = V(l_1-a_1,…,l_k-a_k)$ is a variety, called a linear subspace of $\mathbb{C}^n$ of dimension $n-k$.
I think, it is true when $(a_1,…,a_k)=(0,…,0)$ according to linear algebra, but when $(a_1,…,a_k) \neq (0,…,0)$, $X$ is not a linear subspace because $X$ has not $(0,…,0)$. and I think X is a translate of a subspace of $\mathbb{C}^n$.
Is my opinion true? or What is the author's intention? thanks.
Best Answer
You are thinking of $\mathbb{C}^n$ as a vector space, the author is thinking of it as an affine space. An affine space carries an action of a vector space (translation). An linear subspace of an affine space is one that is an orbit for a sub-vector space of this vector space.