I am self-studying Linear Algebra by Hoffman & Kunze.
Exercise 6 in Section 1.2: "Prove that if two homogenous systems of linear equations in two unknowns have the same solutions, then they are equivalent."
I am trying to solve this problem by induction (see the first bullet point below).
I am hoping to solve this problem using only the information provided in the book thus far. In this text, two systems are equivalent if each equation in one system can be written as a linear combination of equations in the other system. So far the text has defined the following phrases: solution of a linear system of equations, equivalent systems, and homogenous system of equations. It has not yet covered linear dependence.
Some notes on previously asked questions:
- This question seems to be asking the same thing, and while the OP was able to solve it with the hints given in the comments, they did not post the solution. I am trying to solve this problem by induction on the number of equations in one of the systems of equations, as suggested in the comments of that post. My trouble arises when attempting to invoke the inductive hypothesis (IH). Starting with $N+1$ equations, I cannot simply add two equations together to obtain an equivalent system of $N$ equations in order to invoke the IH. The new system of $N$ equations may have solutions that the system with $N+1$ did not.
- This post asks the same question, but the answer uses material and intuition not yet covered by the text at this point.
- I have found this question answered on this blog, but with a different approach.