Let A Rmn and let x be a solution of the least squares problem Ax = b. Show that a vector y Rn will also be a solution if and only if y = x + z, for some vector z N(A). [Hint: N(AT A) = N(A).]

M303 Section 1.8 Notes- Introduction to Linear Maps/Transformations 9-19-16 n If A is m×n matrix, them for any vector xϵR , mulmiplication by A produces new vector A x ϵR ; if we regard vectors in R as inputs on which A acts by multiplication to give output iR m , and we arrive at notion of a function Function/map n m - rule which assigns unique output m to each n T:R → R T(x)ϵR input xϵR n m R R goes from domain to ta