Null space of this matrix is no solution?

Question

When I set this row reduced matrix (which I row reduced using matlab) equal to zero for finding the null space, am I supposed to get no solution? Because x8, the last vector will be equal to zero.

Answer 1

The null-space of a matrix will always contain the zero vector, so we'll never have a "no solution" issue when trying to find the null-space.

Here, to calculate the null-space is to determine the set of vectors $[x_1\:\cdots\:x_8]^T$ such that $$\left[\begin{array}{cccccccc}1 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 2 & 3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\\x_5\\x_6\\x_7\\x_8\end{array}\right]=\left[\begin{array}{c}0\\0\\0\\0\\0\\0\\0\\0\\0\end{array}\right].$$ Converting this into equivalent equations, we must solve the system $$\begin{cases}0=x_1+2x_2\\0=x_3+2x_4+3x_5\\0=x_6+2x_7\\0=x_8\\0=0\\0=0\\0=0\\0=0\\0=0.\end{cases}$$ All the $0=0$ equations mean nothing to the system, so we really only need $$\begin{cases}0=x_1+2x_2\\0=x_3+2x_4+3x_5\\0=x_6+2x_7\\0=x_8,\end{cases}$$ or $$\begin{cases}x_1=-2x_2\\x_3=-2x_4-3x_5\\x_6=-2x_7\\x_8=0.\end{cases}$$ Returning to vector notation, the vectors in our null-space have form $$\left[\begin{array}{c}-2x_2\\x_2\\-2x_4-3x_5\\x_4\\x_5\\-2x_7\\x_7\\0\end{array}\right]=x_2\left[\begin{array}{c}2\\1\\0\\0\\0\\0\\0\\0\end{array}\right]+x_4\left[\begin{array}{c}0\\0\\-2\\1\\0\\0\\0\\0\end{array}\right]+x_5\left[\begin{array}{c}0\\0\\-3\\0\\1\\0\\0\\0\end{array}\right]+x_7\left[\begin{array}{c}0\\0\\0\\0\\0\\-2\\1\\0\end{array}\right],$$ and so our null-space is the space spanned by $$\left[\begin{array}{c}2\\1\\0\\0\\0\\0\\0\\0\end{array}\right],\,\left[\begin{array}{c}0\\0\\-2\\1\\0\\0\\0\\0\end{array}\right],\,\left[\begin{array}{c}0\\0\\-3\\0\\1\\0\\0\\0\end{array}\right],\,\left[\begin{array}{c}0\\0\\0\\0\\0\\-2\\1\\0\end{array}\right].$$