I appreciate any help or answer to my question which is how can we find a basis for an affine subspace "if we can say that"? In other words, for example, if we have a line in $\mathbb{R}^3$ then ofcourse we can write its parametric equation but how can we find its basis? or if we have a plane in $\mathbb{R}^4$, how can we find "basis" which span the whole vectors of this plane? I am deliberately choosing a line in $\mathbb{R}^3$ and a plane in $\mathbb{R}^4$ so that they are not a hyperplane but they are affine subspaces. It is easy to find the basis for subspaces but what we can say when they are affine subspaces?Also, I know that the affine subspace is the translation of a subspace with some vector $a$, that is the affine subspace
\begin{equation*}
A=a+S
\end{equation*}
is the traslation of the subspace $S$ by the vector $a$ for some vector $a\in A$ but I am not able set the argument for finding the basis for $A$. Any help, answers, explantions will be appreciated.
Best Answer
The basis for an affine subspace is of a different form than that of a linear subspace. For a linear subspace $\mathcal{S}$, we define vectors $b_1,b_2,\ldots,b_n$ such that any vector $x \in \mathcal{S} $ can be written as
$$x = \sum\limits_{k=0}^n\alpha_k b_k.$$
However, for an affine subspace $\mathcal{S}_a$, we define the basis $v_1,v_2,\ldots,v_n$ such that any vector $y \in \mathcal{S}_a$, we write $$y = a+ \sum\limits_{k=0}^{n}\beta_k v_k.$$ where $a$ is a translation vector, like what you have mentioned.
Another way of looking at an affine subspace is as a set $\mathcal{S}_a=\left\{x \ | Ax = b\right\}$ and a linear subspace as $\mathcal{S}=\left\{x \ | Ax = 0\right\}$. The matrix $A$ and the vector $b$ completely describe the affine subspace.
Unlike a linear subspace, an affine subspace cannot be written as a linear combination of a bunch of basis vectors since the $\mathbf{0}$ vector does not belong to the affine subspace and an arbitrary linear combination will include the zero vector too.