Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra

Linear algebra’s payoff is practical: many real problems reduce to solving linear systems, and the geometry of a matrix determines whether solutions exist, whether they’re unique, and what the solution set looks like. A linear system can be written as a single matrix equation, Ax = V, where A encodes the coefficients, x holds the unknowns, and V is the target vector. Geometrically, multiplying by A is a linear transformation: solving Ax = V means finding the vector x that lands exactly on V after the transformation reshapes space. This geometric lens matters because it turns algebraic questions—like “does an inverse exist?”—into questions about how A squashes or stretches space.

When A has a non-zero determinant, it does not collapse space into a lower dimension. In that case, there is exactly one vector x that maps to V, and the solution can be obtained by reversing the transformation. The reverse operation is the inverse matrix, A^{-1}, defined by the property A^{-1}A = I, where I is the identity transformation that leaves i-hat and j-hat unmoved (and thus has columns (1,0) and (0,1) in 2D). Examples make the idea concrete: a 90-degree counterclockwise rotation is undone by a 90-degree clockwise rotation; a shear that shifts j-hat right by one unit is undone by a shear that shifts j-hat left by one unit. In higher dimensions, the same principle holds: if the determinant is non-zero, the transformation is invertible, and when the number of equations matches the number of unknowns, the system typically has a unique solution.

Determinant zero changes everything. A transformation with determinant zero squishes space into a lower-dimensional object—like a line, plane, or point—so it cannot be “unsquished” by any function that maps each input vector to a single output vector. For square systems (same number of equations and unknowns), this means no inverse exists. Still, solutions might exist even without an inverse: they occur only when V lies within the collapsed output region. That leads to a more refined classification than “determinant zero” alone.

The key refinement is rank, defined as the number of dimensions in the transformation’s output. A rank-one transformation collapses everything onto a line; rank-two collapses onto a plane; rank-three fills 3D space. The set of all possible outputs of A is the column space, which equals the span of A’s columns—because each column shows where a basis vector lands under the transformation. Rank is therefore the dimension of the column space, and a matrix is full rank when this dimension equals the number of columns.

Finally, the null space (kernel) captures the vectors that collapse to the origin: all x such that Ax = 0. Because linear transformations always keep the origin fixed, the zero vector always belongs to the null space. When A is not full rank, entire subspaces collapse to zero: in 2D, a line of vectors may map to the origin; in 3D, a plane or line may do so depending on how much collapse occurs. In the language of linear systems, when V = 0, the null space describes all solutions—so it explains not just whether solutions exist, but how many and in what geometric form.