Question

let A be 3x3 matrix with (2,0,-1) as first row (4,0,-2) as second row and (0,0,0) as third row .let T be multiplication by A find basis for range of T

Answer 1

If a transformation can be expressed as a matrix, A, then a basis for the image of T lies in A's column space. Since T:R3->R3 you'll require 3 vectors in your basis set if T's image spans the entire codomain. Your basis might not have three vector elements if A's columns contain the zero vector or are linearly dependent (the former is equivalent to the latter really).