How to dynamically fill empty space in a column?

How to find column, row, and null space of a matrix?

• I am very confused between finding the basis of a space, and then just finding the space. Right now with my knowledge, I think that finding a column space is just listing the columns of a matrix, then the basis is the linearly independent vectors of that space. When I find the null space, how do I find the basis? Do RRE to them? Row space, any nonzero row would be in the row space? Can there be a basis for the row space? I have my linear algebra exam soon and I just want to make absolutely sure that I know the specifications. Thank you. : )

• Answer:

  For the column space, yes, you can start with the column and search for the biggest set of independent ones; these will form the basis of your column space. Exactly the same with the row space: find the independent ones, they will form the basis of the row space. But how does one find these independent row and columns, e.g. if you have a big matrix? A better way is to do the Gauss-Jordan elimination: the non-null rows you end up with are a basis of row space... ... and as a bonus, the columns with the pivots after this elimination identify which columns are independent in the original matrix (row operations do not alter the linear combinations of the columns, i.e. a non-zero linear combination of the columns will stay non-zero after any row operation). (Of course, the column and row spaces will have the same dimension, the rank of the matrix.) Interestingly, the same row elimination above can be used to find the null space basis. See below for some details and small examples. .