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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.

sunspots
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HaakonA
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2 Answers2

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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 $T$.

Lorenzo Pompili
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  • Yes, I made a typo. Thank you for your reply. I do not understand how it can be a line, maybe this is the root of my confusion. The elements in T are vectors in $\mathbb{R}^2$. How can the affine subspace be a line? – HaakonA Jun 30 '21 at 10:26
  • $w−v \in \mathbb{R}^2$ is just an element, a vector. Then the generated $\mathrm{span}(w−v)$ space has dimension one. Imagine taking in the plane just the $x$-axis. Now, to this line translate every point in $w$ direction. This is performed coordinate by coordinate. That is how you draw this line. It does not have to contain the zero, since the affine subspace is not a vector subspace for $w\neq 0$. – rarc Jun 30 '21 at 11:16
  • Ok, I think I understand it. So the vectors in T are all the vectors that point to a point on the line representing T. – HaakonA Jun 30 '21 at 11:33
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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.

sunspots
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  • Thank you for the detailed reply. I still struggle a bit with the interpretation and seeing the two-dimensional vectors in T as points on a line. However, I think I get it. You can think about the elements in T as vectors whose coordinates fulfill these equations and thus point to points on this line. – HaakonA Jun 30 '21 at 11:36
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    At the end of the day, vectors in $\mathbb R^n$ are lists of $n$ numbers. And points in $\mathbb R^n$ are lists of $n$ numbers too. So, it's not so strange to use interchangeably vectors and points. You can see a point in $\mathbb R^n$ as a vector that goes from the origin to that point, and viceversa you can see a vector just as the point right at the beginning of its arrow. If you still want to see vectors as arrows, you can do it, and the drawing of the affine space should be something like a set of arrows whose tails are all at the origin and the heads all lie along a line – Lorenzo Pompili Jun 30 '21 at 11:47
  • @HaakonA now that you have the two lines, you can connect to the following question: https://math.stackexchange.com/questions/22116/parametrization-of-a-line – sunspots Jun 30 '21 at 11:48
  • Yeah I get it now. I appreciate the explanation. I was to caught up in the idea that the elements are vectors and therefore must be drawn as vectors. This is not a good mentality to have. – HaakonA Jun 30 '21 at 11:57
  • Don't worry. Sometimes it's better to think of vectors in one way, sometimes in the other one. It is just good to know both :) – Lorenzo Pompili Jun 30 '21 at 12:24