Question

that is Give an example of a linear transformation T: R2X2 → Ral surjective (onto), however is not one-to-one. O No linear transformation, T, exists which satisfies these conditions. There exist such a linear transformation, T. For example: sat

Answer 1

Given: T: R²*2 R [n] is a linear transformation. Let A be the standard matrix of T The dimension of R²*2 4 { R2X2 is the vector space of all 2x2 matrices with real entries, The dimension of R₂ (n) = 4 {Row is the vector space of all . Ipolynomials of degree 3 or less, with . A is a 4x4 matrix Treal colfficients ginen, Tis surjective Conto) Rank of A=4 .. By the rank-mullity theorem: rank of Athullity of A= number of column in A → 4 + hullity of A=4 nullity of A=4-4 nullity of A=0 ie dimension of kernel (T) = 0 Kernel (T) = {o} : T is one-to-one A linear transformation Tis one-to-one, if and only if Kernel (t) = {5} The correct answer is • No linear transformation, T, exist which satisfies these conditions.