Finding Kernels

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding the Kernel of a Linear Transformation Find by solving , then read off a basis and the nullity. The linear transformation is given by , where . Find the kernel : give a basis for it and state its dimension (the nullity). The kernel is a line through the origin in Every solution of is a scalar multiple of , so the kernel is a one-dimensional line through the origin. Nicely done. You found the kernel of a linear transformation by solving a homogeneous system, then read a basis straight from the reduced matrix. Kernel = null space: , so finding it means solving . Row reduce: reduce A to RREF ; pivot columns give basic variables, pivot-free columns give free variables. Read off a basis: with x_3=t free, , so and the nullity is 1 . Always check: a basis vector must satisfy — here . The nullity (here 1 ) measures how much the transformation collapses. With 3 columns and rank 2 , the Rank–Nullity Theorem gives .

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