## The Euclidean plane

The vector space $$\mathbb{R}^2$$ is often depicted by a 2-dimensional plane with two perpendicular axes. The horizontal axis, labelled $$x_1$$, represents the values of the first entry of tuples and the vertical axis, labelled $$x_2$$, represents the values of the second entry of tuples.

Every $$u \in \mathbb{R}^2$$ can represented as a point on the plane as follows: Draw a vertical line crossing the $$x_1$$-axis at the value $$u_1$$ and a horizontal line crossing the $$x_2$$-axis at the value $$u_2$$. The intersection of these two lines is the point on the plane that represents the tuple $$u$$. (For convenience, we often refer to the point as the tuple $$u$$ itself.)

The intersection of the two axes is $$\begin{bmatrix} 0 \\ 0\end{bmatrix}$$ and is called the origin.

The setup above with tuples represented as points forms the basis for 2-dimensional coordinate geometry. However, in the context of vector spaces, the vectors in $$\mathbb{R}^2$$ are normally represented by arrows rather than points. For example, the vector $$u \in \mathbb{R}^2$$ is represented by an arrow with the tail at the origin and the head at the point $$u$$. The figure below illustrates the vectors $$u = \begin{bmatrix} -1 \\ 3\end{bmatrix}$$ and $$v = \begin{bmatrix} 2\\-1\end{bmatrix}$$.

Using arrows, one can work with the vector space $$\mathbb{R}^2$$ purely geometrically. To add $$u$$ and $$v$$, one simply slide (without any rotation) the arrow representing $$v$$ so that its tail coincides with the head of $$u$$. Then the arrow from origin to the head of the slided arrow represents $$u + v$$. Note that one can get the same arrow by sliding the arrow representing $$u$$ so that its tail coincides with the head of $$v$$. This is to be expected since in $$\mathbb{R}^2$$, $$u + v = v + u$$.

To multiply $$u$$ by a scalar $$\alpha$$, if $$\alpha \geq 0$$, create an arrow with the tail at the origin pointing in the same direction as $$u$$ and with length $$\alpha$$ times the length of $$u$$. Hence, $$2u$$ will be an arrow twice as long as $$u$$ pointing in the same direction as $$u$$.

If $$\alpha \lt 0$$, create with the tail at the origin pointing in the opposite direction as $$u$$ and with length $$\alpha$$ times the length of $$u$$.

The figure below illustrates $$u+v$$ and $$-0.5v$$ for $$u = \begin{bmatrix} -1\\3\end{bmatrix}$$ and $$v = \begin{bmatrix} 2 \\ -1\end{bmatrix}$$.

## What about $$\mathbb{R}^3$$?

One can also represent vectors $$\mathbb{R}^3$$ as arrows in a 3-dimensional space. The details are similar to the case for $$\mathbb{R}^2$$ and are left as an exercise.

## Exercise

1. Give a sketch of the span of $$\left \{\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}\right \}$$ in $$\mathbb{R}^3$$.

2. For each of the following, determine if it is a subspace of $$\mathbb{R}^2$$. Explain your answer. (Hint: Try to visualize these sets.)

1. $$\left\{ \begin{bmatrix} x_1\\x_2 \end{bmatrix} : \lvert x_1 \rvert + \lvert x_2 \rvert = 0 \right\}$$

2. $$\left\{ \begin{bmatrix} x_1\\x_2 \end{bmatrix} : 5x_1+3x_2 = 0 \right\}$$

3. $$\left\{ \begin{bmatrix} x_1\\x_2 \end{bmatrix} : x_1^2 \geq x_2 \right\}$$

4. $$\left\{ \begin{bmatrix} x_1\\x_2 \end{bmatrix} : x_1 - x_1x_2 = 0 \right\}$$