I am currently reading through "An Introduction to Convex Polytopes" and to reify some of the concepts in the first chapter I am attempting to construct concrete examples in 3-dimensions so that I have something to visualize for tackling to higher level concepts. One that I am not sure if I have done correctly is the Linear Hull (also known as linear span).
Here is the definition given in the textbook by Bronsted:
Linear Hull:
For any $M \subset \mathbb{R}^{d}$ the linear hull $\operatorname{span}{(M)}$ is the set of all linear combinations of vectors from $M$.
So, I considered a simple example consisting of the following vectors:
$x_1 = \langle -2, 1, 1 \rangle, x_2 = \langle 3, -1, -1 \rangle, x_3 = \langle 2, 0, 1 \rangle$.
The equation of the planes for which linear combinations of each two vectors generates in $P_{x_{i}x_{j}}$ form are as follows:
$$P_{x_1 x_2} = y -z, P_{x_2 x_3}=-x -5y + 2z, P_{x_1 x_3}=x+4y-2z $$
We can plot this in Mathematica as follows:
Plot3D[{y, (x + 5 y)/2, (x + 4 y)/2}, {x, -10, 10}, {y, -10, 10}]
So, is the linear hull of this subset $M$ of $\mathbb{R}^{d}$, the set of all vectors that lie in one of the three planes I listed? What about linear combinations of all three vectors simultaneously? What vectors can't be listed if we allow linear combinations of all three vectors simultaneously, and is that what the definition of a linear hull entails?
Best Answer
The phrase "the set of all linear combinations of vectors from $M$" refers to the set of all vectors of the form
$$a_1x_1 + a_2x_2 + \cdots + a_kx_k$$ where $k$ is a nonnegative integer, $a_1,\ldots,a_k\in\mathbb{R}$ and $x_1,\ldots,x_k\in M$.
For example, the span of $\{x_1,x_2,x_3\}$ is defined to be the set of all vectors expressible in the form
$$a_1x_1 + a_2x_2 + a_3x_3,$$ for $a_1,a_2,a_3\in\mathbb{R}.$
In the particular example of $x_1,x_2,x_3$ that you gave, notice that for any $a,b,c\in\mathbb{R}$, some computation shows that
$$\langle a,b,c\rangle = (a+5b-2c)x_1 + (a+4b-2c)x_2 + (c-b)x_3$$
and so every point $\langle a,b,c\rangle\in\mathbb{R}^3$ is in the linear span of this set. That is, in this example, $\mathrm{span}\{x_1,x_2,x_3\} = \mathbb{R}^3$.