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Font Styles Paragraph Definition 1: Given La linear transformation from a vector space V into itself,...


Font Styles Paragraph Definition 1: Given La linear transformation from a vector space V into itself, we say that is diagonal
it way Given the linian transformation Li Pelt) - Palt) defined by Llat² tbt + c)=ct² tbt ta 1) Find the matrix A representin
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
it way Given the linian transformation Li Pelt) - Palt) defined by Llat² tbt + c)=ct² tbt ta 1) Find the matrix A representing L with respect to the basis s ={pt2, 8th 2) find the eigenvalues of L 3) find the eigenvectors oft 4) show that I is diagonalisable 5) Find a diagonal matrix D and an orthogonal matrix P such that D=P"AP 6) find an orthernormal bases for P₂ (t) relevant to which the linear transformation! is represented by the matrix pobtained in part 5
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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=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 = sum3) 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 RI4 ng algebraic multiplicity of every matrix is equal geometric multiplicity diagonalizable to So, A is or if we take pe 0 the

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