For a 3x3 matrix whose columns span only a plane, what does its nullspace say about solutions of Ax = b?

## Setting the scene A is 3 x 3, so it maps R^3 to R^3. Suppose its three columns are not independent: they span a **plane** through the origin rather than all of R^3. Then rank(A) = 2, the matrix is rank deficient, and it is singular - no inverse exists. ## The column space controls existence The product Ax is, by definition, a weighted combination of A's columns with weights taken from x. So the set of all reachable outputs is precisely the span of the columns: the **column space**. If that span is a plane, the reachable set is that plane and nothing else. Therefore Ax = b has a solution if and only if b lies in the plane. A b sitting off the plane, even slightly, is simply not in the range of the map. This is the existence half of the story, and it is entirely a statement about the output side, R^3 as a destination. ## The nullspace controls uniqueness Rank-nullity says rank + nullity = number of columns = 3, so nullity = 3 - 2 = 1. There is a non-zero vector v with Av = 0, and every multiple of it also maps to zero. The nullspace is a **line** through the origin in R^3 - the direction the matrix crushes. Now suppose some x_p satisfies A x_p = b. For any scalar t, ``` A(x_p + t*v) = A x_p + t*(A v) = b + t*0 = b ``` so x_p + t*v is also a solution. Conversely, if x1 and x2 both solve the system, then A(x1 - x2) = b - b = 0, so their difference lies in the nullspace. Put together: ``` solution set = x_p + null(A) = { x_p + t*v : t real } ``` The solution set is a line: a shifted copy of the nullspace line, translated by any one particular solution. It passes through the origin only when b = 0. ## The full picture in three cases - **b off the plane:** no solutions. The system is inconsistent. - **b in the plane, b non-zero:** infinitely many solutions, forming a line that does not pass through the origin. - **b = 0:** infinitely many solutions, forming the nullspace line itself, which does pass through the origin. The zero vector is always among them, so a nullspace is never empty - it always contains 0, even for a full-rank matrix, where it contains nothing else. Notice that a square matrix guarantees nothing. 'Square' is not 'invertible'; only full rank is. ## Row space, and where the nullspace points Row rank equals column rank, so the row space here is also two-dimensional: the three rows span a plane in R^3 and one row is a combination of the other two. There is a clean relationship: the nullspace is exactly the set of vectors orthogonal to every row, since each entry of Ax is the dot product of a row with x, and Ax = 0 means all those dot products vanish. So the nullspace line is perpendicular to the row-space plane. In R^3 a plane and its perpendicular line account for 2 + 1 = 3 dimensions - rank-nullity again, seen geometrically. ## Reading it as a diagnosis When a linear system built from real data turns out to have this structure, the interesting question is *why* a direction collapsed. A rank-deficient matrix means some combination of the input coordinates makes no difference to the output at all: moving along v changes the inputs but leaves Ax untouched. Any procedure that reports 'the' answer is really reporting one arbitrary point on a line of equally valid answers, chosen by whatever tie-break the procedure happens to apply. That is why the honest statement to a colleague is not 'the computation is unstable' but 'the problem as posed does not determine a unique answer; here is the direction along which the answer is free.' Naming v tells everyone exactly which combination of inputs is unidentifiable, which usually points straight at how the matrix was constructed. ## The summary to be able to recite Column space answers existence, nullspace answers uniqueness, and rank-nullity connects their sizes. For a rank-2 3x3 matrix: reachable outputs form a plane, unreachable b gives no solution, and every consistent b gives a one-parameter line of solutions.