I figured out the solution with 2 independent variables and 2 free variables
The solution set is: $$[x,y,z,w]=[10,23,0,0]+t[-1,1,1,0]+s[3/2,11/2,0,1]$$
However, how to give a geometric interpretation for this solution? Thanks
2026-04-25 21:51:08.1777153868
How to give a geometric interpretation of linear system which has four vectors?
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The set you are referring to is the linear span of two linearly independent vectors, hence it is of dimension $2$, in a space of dimension $4$. Such an object is what we would describe as a $4$-line geometrically, which means it is a $2$ dimensional object embedded in a space of dimension $4$. It's relation with respect to space, will be the same as the relation between a normal line and $3$ dimensions, in that it will extend in all basis directions except two. So it would be a plane, extending in two directions, but it would leave two directions out instead of one.
In general, an $m-2$ dimensional object embedded in $m$ dimensional space is called an $m$-line.