vectors pure and applied. exercise 6.4.2 OIK IIC rather than Example 6.4.1 Let ul, u2 be a basis for F2. The linear map β : F., p given by is non-diagonalisable. hat β is diagonali able with resp...
OIK IIC rather than Example 6.4.1 Let ul, u2 be a basis for F2. The linear map β : F., p given by is non-diagonalisable. hat β is diagonali able with respect to some basis. Then β would have Proof Suppose t matrix representation D=(d, 0 say, with respect to that basis and ß2 would have matrix representation 2 (d2 0 with respect to that basis. However for all xj, so β-0 and β2 has matrix representation with respect to every basis. We deduce that d-d-0, so di = d2 = 0 and β = 0 which is absurd. Thus β is not diagonalisable. Exercise 6.4.2 Here is a slightly different proof that the mapping β of Example 6.4.1 is not diagonalisable. (i) Find the characteristic polynomial o β and show that() İs the only eigenvalue of β. (ii) Find all the eigenvectors of β and show that they do not span F". Fortunately, the map just given is the 'typical' non-diagonalisable linear map for C
OIK IIC rather than Example 6.4.1 Let ul, u2 be a basis for F2. The linear map β : F., p given by is non-diagonalisable. hat β is diagonali able with respect to some basis. Then β would have Proof Suppose t matrix representation D=(d, 0 say, with respect to that basis and ß2 would have matrix representation 2 (d2 0 with respect to that basis. However for all xj, so β-0 and β2 has matrix representation with respect to every basis. We deduce that d-d-0, so di = d2 = 0 and β = 0 which is absurd. Thus β is not diagonalisable. Exercise 6.4.2 Here is a slightly different proof that the mapping β of Example 6.4.1 is not diagonalisable. (i) Find the characteristic polynomial o β and show that() İs the only eigenvalue of β. (ii) Find all the eigenvectors of β and show that they do not span F". Fortunately, the map just given is the 'typical' non-diagonalisable linear map for C