I came across the following multiple choice question:
The number of linearly independent solution of the homogeneous system of linear equations $AX=0$, where $X$ consists of $n$ unknowns and $A$ consists of $m$ linearly independent rows is
$(A)$ $m-n$ $\space$$(B)$ $m$ $\space$$(C)$ $n-m$ $\space$$(D)$ none of these
I think the answer will be $(D)$ because:
When $m=n$, in this case it would mean a square matrix with all linearly independent rows, which implies unique solution. When $m<n$, it would mean that the rank of the matrix is less than number of unknowns in the system, which would mean infinite solutions and these solutions must be linearly dependent (Am I going right?). When $m>n$, there will be no solution (I am not sure about this one)
I think there is something wrong about my answer because I remember something like: number of linearly independent solutions = number of unknowns – rank, from my Linear Algebra class. But I am not sure how to relate to it here..
Thanks..
Best Answer
When $m=n$, we get unique solution that is trivial and hence linearly dependent. So, we get $0$ linearly independent solution (Here, $n-m=0$).
When $m<n$, we also get non-trivial solution along with the trivial solution. We discard the trivial solution because we want only linearly independent solutions. Since we have $m$ linearly independent rows, the rank of $m \times n$ matrix is $m$. Hence $n-m$ linearly independent solutions
The number of linearly independent rows is equal to number of linearly independent columns. Thus, number of linearly independent rows cannot exceed the number of columns. So, $m>n$ is an impossible case.
The correct option is $(C)$ $n-m$.