The Null Space as Kernel

Linear Algebra (Matrices) · Axiom Academy

LESSON The Null Space as Kernel The kernel of T is exactly the null space of its matrix — the directions T collapses to zero. 1. The Kernel Is What Collapses to Zero For a linear transformation , the kernel is the set of inputs that land on the zero vector. Because is the matrix equation defining the null space, the kernel and the null space are the same set . Watch the singular matrix act on the plane. Vectors along the dashed direction get squashed straight onto the origin — those are the kernel. Other vectors move but stay nonzero — they survive. The kernel is never a stray scatter of points — it is a subspace of the domain. For our matrix the kernel is the entire line y = -x through the origin. Watch every point of that line map to the single point : scale a kernel vector and it stays in the kernel; add two and the sum stays too. Linearity does all the work. If and are both crushed to zero, then , and . Both results stay in the kernel, so the kernel is closed — a genuine subspace. 3. One-to-One Exactly When the Kernel Is Trivial The size of the kernel decides whether T is injective (one-to-one). On the left, an invertible B collapses nothing but , so distinct inputs keep distinct outputs. On the right, our singular A collapses a whole line, so a whole line of different inputs all share the output — not one-to-one. Watch both grids land. . Only maps to ; different inputs give different outputs. is the line y=-x . A whole line of inputs collapses to — outputs collide.

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