So I'm a long distance student and I need some help to bounce ideas off of other people who understand the work. Fellow students are few and far between. So while this is an assignment question, I just need help to check if my logic and understanding is correct
Question
Suppose $A$ is an $m \times n$ matrix where $m < n$: Which of the following statements is/are true?
A. The non-homogeneous system $Ax =b$ has at least one solution.
B. The homogeneous system $Ax = 0$ has a unique solution.
Answer
A: I don't think A is correct. $Ax=b$ will only have at least one solution IF the linear system is consistent. In our study guide it said that if $m < n$ then infinitely many solutions exist. But I think this is a trick question.
B: True, homogenous systems always has the trivial solution which is a unique solution.
Best Answer
Both are false. For example, consider $m = 3$ and $n = 4$.
For part A, consider: $$ A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \qquad\text{and}\qquad b = \begin{bmatrix} 0 \\ 0 \\ 7 \end{bmatrix} $$ Since the third row of $A$ is all zeroes, any linear combinations of its columns will give us a $0$ in the third entry (and not $7$).
For part B, use the same matrix $A$ from part A and observe that (besides the trivial solution), another solution to the homogeneous system is: $$ x = \begin{bmatrix} 0 \\ 0 \\ 8 \\ 9 \end{bmatrix} $$