Question

if you have matrix, how do you go about determining the basis for a)the row space and b) column space?

Answer 1

I sometimes get this confused. But I believe if you reduce the matrix and take each of the independent columns and put them into vector form you have the basis for the column space of the matrix. I think if you transpose the matrix and take the independent columns (they were the rows) then you will have the basis for the row space (this is where I get unsure) The idea is that all of the rows and columns can be made up from some combination of scalars and those vectors.

Answer 2

thanks. ill give it a shot

Answer 3

I get these confused as well. I know that to determine the basis of the Null and Column spaces you must row reduce and the span of the Null space is determined by the free variables, while the span of the column space is determined by the pivot rows of the original matrix, determined by row reduction