i am having difficulties with a few questions about ranks
in a mxn matrix A prove dim col A + dim nul A^T = m
starting with the rank theorem rank A + dim nul A = n
by the definition of rank
rank A = dim col A
dim col A + dim nul A = n
not really sure how to go any further in this one to prove the given statement. i also know col A^T is equivelent to row A but i dont know how this would help.
another question i have is given a mxn matrix A and b belonging to R^m what has to be true about rank [A b] and rank A in order for the equation Ax=b to be consistent.
rank A is the dim col A so i think that rank A and rank[A b] would have to be equal for the system Ax=b to be consistent?
in a mxn matrix A prove dim col A + dim nul A^T = m
starting with the rank theorem rank A + dim nul A = n
by the definition of rank
rank A = dim col A
dim col A + dim nul A = n
not really sure how to go any further in this one to prove the given statement. i also know col A^T is equivelent to row A but i dont know how this would help.
another question i have is given a mxn matrix A and b belonging to R^m what has to be true about rank [A b] and rank A in order for the equation Ax=b to be consistent.
rank A is the dim col A so i think that rank A and rank[A b] would have to be equal for the system Ax=b to be consistent?