A Matrix Splits Space into Four Subspaces

Drag the input vector x. This matrix A is a flat one whose two columns point the same way (det = 0), so wherever you drag, the output A x lands only on a single line. That line is the column space. Why? Because A x is some amount of the first column plus some amount of the second, a combination of the two columns, and since the columns are parallel, all that combination can make is that one parallel line. The output can never leave that line. Every reachable result is trapped on this one-dimensional line. This is the image, the column space.

On the same flat matrix, now look at the input side. There are inputs that die to zero. Drag x and you find that when you place it on a certain line, the output A x sticks exactly to the origin. That line is the null space. Inputs arriving along that direction are crushed entirely to zero, leaving no trace. Stray just a little off that line and the output jumps right up onto the column space. Not a single point but a whole line maps to zero. That is why A x = 0 has not just x = 0 but infinitely many solutions. This very fact, that the null space is nonzero, becomes the key to why solutions can come in multitudes.

Now use the slider to collapse the matrix from alive to flat. With the slider at one end, A is full rank 2: the unit square turns into a healthy parallelogram, and the only input going to zero is x = 0, so the nullity is 0. Turn the slider so one column tips toward the other and the two columns become parallel (det goes to 0); the parallelogram collapses into a single segment. At that instant the rank is 1 and the nullity is 1, because a whole line now goes to zero. Either way, rank plus nullity is always 2: the input dimension is split exactly between the two children. The surviving dimensions (rank) and the dead dimensions (nullity) always sum to the incoming dimension n. This is the rank-nullity theorem.

Look at the input plane again. The null space is not alone there: the row space lives on the input side too. The row space is the directions the matrix's two rows span, and on a flat matrix it is also a single line. Toggle the two lines on and you see the row space and the null space meet at exactly 90 degrees. The two are a perpendicular pair that complement each other. Why? Because each component of A x is the dot product of a row with x, and if x lies in the null space that dot product is 0, meaning x is perpendicular to every row. So every input vector splits into two pieces: the row-space part survives in the output, the null-space part vanishes to zero. The half that lives and the half that dies are split at a right angle.

Now solve A x = b with the four-subspace picture. Use the slider to move the target b. If b does not lie on the column-space line, there is no input that reaches it at all: zero solutions. Raise b onto the column space and solutions suddenly appear, and not just one: a single particular solution plus the entire null space, that is, a whole line shifted along the null-space direction, are all solutions. Infinitely many. The particular solution is one input that reaches b, and adding any of the directions that die to zero leaves the output still at b. Had the matrix been full rank with only 0 in its null space, the solution would be exactly one, the answer the inverse gave. Zero, one, or infinitely many: all three branches come straight from the shape of the four subspaces.