Question

Font Styles Paragraph Definition 1: Given La linear transformation from a vector space V into itself, we say that is diagonalizable iff there exists a basis S relevant to which can be represented by a diagonal matrix D. Definition 2: If the matrix A represents the linear transformation L with respect to the basis S, then the eigenvalues of L are the eigenvalues of the matrix A. I Definition 3: If the matrix A represents the linear transformation L with respect to the basis S, then the eigenvectors of A are the coordinate vectors of the eigenvectors of L with respect to the basis S. Definition 4 If the matrix A represents the linear transformation L with respect to the basis 5 And If A is diagonalizable such that D=p-1AP. Then The columns of the matrix P will represent the coordinate vectors of the eigenvectors of L with respect to the basis S and The obtained eigenvectors of L are the elements of a basis T relevant to which Lis represented by the diagonal matrix D. Activate ited States ORA e W

Answer 1

Given! LiP2 (t) Palt defined by Llat²tbt tol ct tbt ta • We have to find matrix A representing t with respect to the basis g={It², 8t it) Now 1197?) = 1 = 0(14%) +008t ) +1:17) 1 (8t) = 8 + = 01772)+1(8t) +0.19) 119) •7+2 1.1972) 10.68+)+0.17) 0 A = 0 U

Next to find the eigen values of l that is of A the characteristic can of A is Siol. t. 9ed-Weo where gia truce A FL S2 = sum of Mincipal ninars of a -160lt lolt 10*1 cm 1354 I det A: 11-1)= -1 .: the characteristic eqn is 12-92 :30 the roots of this quadratic equations are lilni so eigen values of Lave lilil

3) To find eigen vectory of Lie oft il fil del consider (Ad 1) x where x is eigen vector 3 101 00 6 113 HI 0 LOT 0 a REIRA RIXI na J 0 0 0 sitER - HI-H3 = 0 il. Mi = ng let nzit, 12 = 5,7 lit 311 go eigen vector is x - 1 t co مل H1 3 t 50, eigen vectors for del are Y por na-1 consider (A do x = 0 0 NI + ! ت 0 4 2 0 213. +1 consider augmented matrix +1 - RoPRI N R2 - 2 6 -> H170330, 1200 put ug = t tER 11124, 1250, 3=1 + 13 은 so eigen vecfut is To

4 ng algebraic multiplicity of every matrix is equal geometric multiplicity diagonalizable to So, A is or if we take pe 0 then O 0 - PAP D o 06-1 so, A is diagonaliauble (from 4) AP 1 0 6) The orthogonal basis for Ipatinis -tit -