Question

Linear algebra confusion: We haven't learned about rank and such:
For the following I am looking for examples or checking to see why something is impossible.
a.) What does it mean when we have an inconsistent 2x3 linear system
-What if we reversed it to 3x2? Even put other numbers in place.
b.) Can a 2x3 linear system have a unique solution
-What if we reversed it to 3x2? Even put other numbers in place.
c.) Can a 3x2 linear system have infinitely many soluitons
-What if we reversed it to 2x3? Even put other numbers in place.

We had just one day of class and we didn't discuss: inconsistent, unique, infinitely many solutions.
Where can I also reference to learn about this.

Answer 1

Answer-(a). By an inconsistent linear system we mean a system of 2 linear equations in 3 variables for which there is no solution. For examples, if we consider the following system of inconsistent linear equations:

then it does not have a solution.

By an inconsistent linear system we mean a system of 3 linear equations in 3 variables for which there is no solution. For examples, if we consider the following system of inconsistent linear equations:

then it does not have a solution.

(b)&(c). No, a linear system can never (if it is consistent) have a unique solution (it will infact always have infinitely many solutions, if consistent). For example, consider the following consistent system of linear equations:

for which each of elements in the set is solution so the above linear system has infinitely many solutions. A consistent system of linear equation may have unique as well as infinitely many solutions. For example, consider

which clearly has unique solution

Now, consider

which clear has infinitely many solutions for any