Column Space & Row Space

Linear Algebra for Machine Learning · Axiom Academy

Finding Basis via Pivot Columns The column space of a matrix A is the set of all possible linear combinations of the columns of A. It's the range of the transformation defined by A. The column space is a subspace of ℝᵐ where m is the number of rows. Finding Column Space Basis: Worked Example Step 3: Identify Pivot Columns Pivot columns are in positions 1 and 3 (first and third columns). Step 4: Basis for Column Space Use the ORIGINAL columns (not the reduced ones) corresponding to pivots: These two vectors form a basis for the column space. Dimension = 2. We use ORIGINAL matrix columns, not reduced columns Pivot columns from RREF indicate which original columns to use Dimension of column space = rank(A) Column space is also called the range or image of A Difference: Column Space vs Null Space Column Space: All possible outputs of the transformation (where do vectors map to?) Null Space: All vectors that map to zero (what's in the kernel?) Application: Solvability of Systems System Ax = b has a solution if and only if b is in the column space of A. Equivalently: b must be expressible as a linear combination of columns of A.

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.