Finding Images

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding the Image of a Matrix Transformation Find from the pivot columns, get a basis, and decide whether T is onto. Let be the linear transformation , where . Find a basis for the image , state its dimension (the rank of A ), and determine whether T is onto. Nice work! You found the image of a matrix transformation straight from the pivot columns. Image = column space: , the span of the columns of A . Basis = original pivot columns: row-reduce to find the pivots, then take the matching columns of the original A (here columns 1 and 2) — never the reduced columns. Onto check: is onto only if . Here rank =2 , so T is not onto; the image is a plane through the origin in . This ties to the rank–nullity theorem: . Here 2 + 2 = 4 , the dimension of the domain .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.