Kernel & Image of Linear Maps

Linear Algebra for Machine Learning · Axiom Academy

Geometric Interpretation of Null & Column Space Terminology: Kernel = Null Space For a linear transformation T(x) = Ax, the kernel is the same as the null space: it's the set of all inputs that map to zero. Terminology: Image = Column Space The image is the same as the column space: it's the set of all possible outputs of the transformation. The kernel represents the "directions" that the transformation collapses to the origin. These are lost in the transformation. The image is the set of all points you can reach by applying the transformation. It's the effective "output space" of the transformation. Imagine a 3D transformation that projects onto a 2D plane: Kernel: the line perpendicular to the plane (gets squashed) Image: the entire 2D plane (what you can reach) Rank-Nullity Theorem (Geometric Version) Interpretation: The input space is divided into two orthogonal parts: The kernel (null space) of dimension nullity(A) The orthogonal complement of the kernel (row space) of dimension rank(A) A transformation is injective if different inputs map to different outputs. A transformation is surjective if the image equals the entire target space. If A is square, bijective means A is invertible.

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