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Suppose I have a matrix M \begin{bmatrix} 1 & 0 & 0 & 1 \\[0.3em] 0 & 1 & 1 & 1 \end{bmatrix} With its field being $\mathbb{F}_2$. How do I calculate its kernel?

pokemonfan
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  • Find all $\mathbf x = [x_1 x_2 x_3 x_4]^T$ such that $M \mathbf x = \mathbb 0$, keeping in mind that each $x_i$ is either $0$ or $1$ and doing calculations over $\mathbb F_2$. – Théophile Nov 09 '15 at 19:44
  • You can read the kernel directly from a row-reduced echelon form matrix, which you have. See http://math.stackexchange.com/a/1521354/265466 for examples. This is equivalent to solving the equations, but much simpler. – amd Nov 09 '15 at 20:23

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Set $x=\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}$ and look at the equations you get from $Mx=0$.

5xum
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