#linearalgebra find bases and dimensions of the 4 fundamental subspaces

for column space, reduce the given matrix into the form U, then put the pivot columns from the given matrix into the set to obtain basis.

for row space, use the matrix U obtained from elimination then form basis with pivot rows of U (not given matrix!).

for nullspace, consider free variables in the matrix U and compute special solutions accordingly. The vectors you get for the special solutions will form a basis of the nullspace.

for left nullspace, first compute transpose of the given matrix then follow the idea of getting basis for the nullspace.

for dimensions, just count out the number of vectors you get in each basis.