Null Spaces Question
My version of question 1 from the online HW due on 9/24 asks us to find a non-zero vector in the null space of a $2\times 3$ matrix. A sample solution looks like so:
In the written example to help with our homework, it claims to have row reduced the original matrix, but it appears to not be fully in RREF state. In fact, this may just be me but I have no clue how they got to the second matrix. So now I have no clue how to do this problem. I tried getting fully into RREF form and set each resultant equation to zero and solve for the respective variable like we learned in class, but this never grants a correct answer. What am I missing here?
Comments
Well, I agree that their second matrix is not in RREF but it is row equivalent to the first matrix. It looks to me like they did it this way because they want integer solutions. Using the fully reduced form, it's easy to get integer solutions as well. For this example, the RREF is
$$
\left(\begin{array}{rrr}
1 & 0 & \frac{55}{31} \\
0 & 1 & -\frac{15}{31}
\end{array}\right).
$$
From there, it's easy to see that a vector in the null-space is
$$
\left(\begin{array}{r}
-\frac{55}{31} \\
\frac{15}{31} \\
1
\end{array}\right).
$$
Of course, any constant multiple of this vector is also in the null-space. The easiest such vector with integer entries is
$$
\left(\begin{array}{r}
-55\\
15 \\
31
\end{array}\right).
$$
You can get their answer by multiplying by 7.