Linear Algebra Answers

(a) Let A be a square matrix and fA(x) (x = lambda) its characteristic polynomial. In each of the following cases (i) to (iv), write down whether
(D) A is diagonalizable over R
(N) A is not diagonalizable over R
(U) it is not possible to say one way or the other.
(i) fA(x) = (x - 3)^2(x - 5)
(ii) fA(x) = (x^2 - 1)(x^2 - 2)
(iii) fA(x) = (x^2 + 6)(x - 1)(X + 2)
(iv) fA(x) = (x^2 + 3)^2
(b) For each case that you marked (N), say whether or not the matrix can
certainly be diagonalized over C

Let A = (-2 -1 -3 6 1 // 1 1 1 -2 1 // 2 3 1 -2 4 ) Find a basis for the column space Col(A) of A, and a basis for the null spaceNul(A) of A.Show that for any matrix A, if u and v are in the null space of A, then
so is u + v.

[b] u[/b] and [b]v[/b] are two non null vectors in [b]R[sub]n[/sub][/b],&
||u|| and ||v|| denote their respective lengths, and.&
||u – v|| denotes difference between u and v.
If ||u||=1 and ||v||=1 and ||u – v|| = ,& & then the u and v are&
& i. orthogonal&
& ii. Linearly independent&
& iii. Both i. and ii.&
& iv. All of them