MAT188H1 Midterm: MAT188H1_20169_641483478629mat188tut10sol

• School: UTSG
• Department: Mathematics
• Course: MAT188H1
• Professor: All

Document Summary

Faculty of applied science & engineering, university of toronto. Tutorial problems 10: three of the four vectors x1 = . 5 as a linear combination of the eigenvectors of a. Solution: notice that b = 3x1 + 2x2 x4. Therefore, a188b = a188(3x1 + 2x2 x4) = 3a188x1 + Now, from part (a), we know that x1, x2, x4 are eigenvectors of a. Av = v, then akv = ak 1(av) = ak 1v = . 2 x2 188: let a = . 1 1 (a) find all eigenvalues of a and a basis for each eigenspace. (b) find an invertible matrix p and a diagonal matrix d such that p 1ap = d and use this to nd a 1. Solution: (a) computing the characteristic polynomial of a, we get (check this!) det( i a) = det . = ( 2)( 1)( + 1) Therefore the eigenvalues of a are = 1, 1, 2.