Find the dimension of a subspace of $\operatorname{Hom}(\mathbb R^3, \mathbb R^3)$

Question

Find the dimension of the subspace $W \subset \operatorname{Hom}(\mathbb R^3, \mathbb R^3)$ (= space of all linear applications between $\mathbb R^3$ and $\mathbb R^3$) defined by $$ W = \{L \in \operatorname{Hom}(\mathbb R^3, \mathbb R^3)\: : \:\operatorname{Im}(L) \subset S\}$$

where $S = \bigl\{\, (x^1, x^2, x^3) \in \mathbb R^3\ :\ x^1 + x^2 + x^3 - 1\leq 0\ ,\ \ x^1 + x^2 + x^3 +1\geq 0 \, \bigr\}$.

I cannot understand how to start this exercise. Some helps?

Thank You

Answer 1

Here is a guideline.

1. Show that for a linear subspace to be contained between two planes of equations $x+y+z=-1$ and $x+y+z=1$, it has to be contained within the plane of equation $x+y+z=0$. It is an equivalence.
2. This means that the range of the linear maps that you are considering has to lie within a $2$ dimensional subspace of $\Bbb R^3$. To build such a linear map, you have to choose independently the image of $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ (these images make the columns of the matrix of the linear map in this basis).
3. You have three independent choices of two parameters (the coordinates of the images in any basis of the target plane), therefore this space of maps is described linearly by $6$ independent parameters and has dimension $6$.