Drawing an affine subspace

Question

Given an affine subspace $T = w + U$, where $U = span(w-v),$ $w = (4,2),$ and $v = (1,1)$. How would one go about graphical representing it? I have made some drawings of the vectors in $T$, but I am not sure if I have drawn the whole of $T$ in a good way. We are in $\mathbb{R}^2$. The question is related to my question about Affine subspace of two vectors in a field K

In general, I am looking for a way to understand how to make these drawings. What I have read so far does not help me to understand the process.

Answer 1

Let us begin with $w-v = (4,2) - (1,1) = (3,1).$ Now, $U = span(w-v) = \{ a(3,1) : a \in \mathbb{R}\}$ is a subspace, so it includes any scalar multiple of $(3,1).$ Thus, if we interpret this geometrically, then $U$ is the set of points that satisfy the following equation $y=\frac{1}{3}x.$

What about $w + U?$ Let us rewrite this as $\{w\} + U,$ which is also known as the coset of $U$ containing $w.$ It is defined such that $\{w\} + U = \{ w + u : u \in U\}.$ Now, let us manipulate the right-hand side such that $\{ w + u : u \in U\} = \{ (4,2) + a(3,1) : a \in \mathbb{R}\}.$ Thus, if we interpret this geometrically, then $w + U$ is the set of points that satisfy the following equation $y=\frac{2}{3} + \frac{1}{3}x.$ In other words, we notice that the subspace and the affine subspace are parallel.

Answer 2

I guess you mean $U=$span$(w-v)$. T should be the line that goes through w and v in this case. In general, affine 1-dimensional subspaces of $\mathbb R^2$ are lines, so you just have to determine two points that belong to it to draw the line. In this case, $T=w+$span$(w-v)$, thus it's an easy check that both $v$ and $w$ belong to $T$.