Line that passes between two vectors

Question

I encountered the following in a text book I'm reading and I can't seem to understand why this is true (I'm translating this into English so excuse me if I'm not using the correct english terms):

Let $x,y\in\mathbb{R}^n $ then the interval between $x$ and $y$ is the group: $\{\lambda x+(1-\lambda)y: 0\leq\lambda\leq1\}$

I've tried this out on a pair over $\mathbb{R}^2$ and it seems to work but I can't see why this is true.

For example, take the two points: $(0,0), (2,0)$ and you get that the interval between these two points can be expressed by: $(2-2\lambda, 0)$

Can anyone tell me why this is true?

Answer 1

The key here is to convince yourself of the following:

• Any line segment can be expressed in the form $u + \lambda v$ for vectors $u,v$ and $\lambda \in [0,1]$.
• $\lambda x+(1-\lambda)y = y + \lambda(x-y)$ is an expression of the above form
• When $\lambda = 0$, the result is $y$
• When $\lambda = 1$, the result is $x$

Answer 2

How would you describe a line segment between two points $x$ and $y$ in $\mathbb{R}^n$? One possibility is to compute the linear equation describing the line using the coordinates of $x$ and $y$. If this equation is say $a^T z = b$, then one description of the line segment is $\{z \in \mathbb{R}^n \ | \ a^T z = b, x \leq z \leq y \}.$ where the inequalities are taken component wise (with a slight abuse of notation since I assume $x_i \leq y_i \ \forall i = 1, \dots, n$.

What you could do then is to compare the two descriptions and show that they are equivalent.