I am Writing a paper on ECC for Linear Algebra and I found a nice resource about Hamming Codes. it gives a nice walk through on the way they work and how to construct them... really good. I have a problem in a few areas though. the walk through just jumps into some aspects with out explanation and I am getting lost in the shuffle. for instance. to warm you up, they give you a nice 3 x 4 binary matrix. the basis for the column space was simple, and so was the basis for the Nul space, in this case a basis of the Nul Space is:         that was self evident. however, they completed the Nul space. they added the zero vector, big surprise :-) , but they also added the vector:     I have no idea how they got that vector. the Nul space is very important in hamming codes, and I would like to know what is wrong that I can not see where this vector came from. thanks for the help.