Let 7: IRa -+ tRB by T(r,y, z,u) : (r * 2y * z,3r * U * 2u,4x* 39 * z * Zta) \\[ (a) Find the matrix of 7 relative to the standard bases for a and 3. 5 (b) What is the rank of 7 and the nullity of 7...

LINEAR ALGEBRA PROJECT 2- LIhIEARTRJ.N SFORMATIONS AND MATRICES (1) -+ Let 7: IRa tRB by 6 : T(r,y, z,u) (r 2y * * z,3r 2u,4x* 39 z Zta) * * * * U \\[ a 3. (a) Find the matrix of 7 relative to the standard bases for and (b) What is the rank of 7 and the nullity of 7 5 tt (c) Give a basis for the kernel of ? and for the image of 7. q (2) Let ? be a linear operator on Y. If .\ is an eigenvalue for ? show that the set : : Vs e V;Tu ,\u} that is I/1 consists of all eigenvectors for .\, is a subspace {u (called the eigenspace. (3) Let T, [/ be two linear operators on a vector space I/. Show that TU and 1 UT have the same eigenvalues. /t 3\ -3 (B) LetA:lB sl 1 -B 4) \o -6 (a) Find the characteristic polynomial for A, chara(t) (b) Find the eigenvalues of A, corresponding eigenvectors and bases for the eigenspaces. - 30( Verify the Cayley-Hamilton Theorem L,pLh.L i") -p3 (4) Prove that if a linear operator T on a space 7 of 1 dimension rz has n distinct eigenvalues then it represented can be by a diagonal matrix. l' 2\ /o o 1 (5) Diagonalize n: thematrix O + 2 { | Ir 0 2/ \0