With the notion of a span, we can expand our techniques for checking the consistency of a linear system. Theorem SLSLC tells us a system is consistent if and only if the vector of constants is a linear combination of the columns of the coefficient matrix. This is because Theorem SLSLC says that any solution to the system will provide a linear combination of the columns of the coefficient that equals the vector of constants. So consistency of a system is equivalent to the membership of the vector of constants in the span of the columns of the coefficient matrix. Read that last sentence again carefully. We will see this idea again, but more formally, in Theorem CSCS.
We will reprise Sage SLC, which is based on Archetype F. We again make use of the matrix method .columns() to get all of the columns into a list at once.
You could try to find an example of a vector of constants which would create an inconsistent system with this coefficient matrix. But there is no such thing. Here is why — the null space of coeff is trivial, just the zero vector.
The system is consistent, as we have shown, so we can apply Theorem PSPHS. We can read Theorem PSPHS as saying any two different solutions of the system will differ by an element of the null space, and the only possibility for this null space vector is just the zero vector. In other words, any two solutions cannot be different.