how to do part C ? (8+2+7+8 points). Let F be a field, not necessarily algebraic

Question

how to do part C ?

(8+2+7+8 points). Let F be a field, not necessarily algebraically closed. For each integer n 1, denote by Mn xn(F) the set of n × n matrices with entries in F. (a) Suppose that A e Mnxn(F satisfies A2-A. Find all possible eigenvalues of A, and prove that A is diagonalisable. Remark. Such an A is called an idempotent matrix. An example in case F and n = 3 is the matrix of the orthogonal projection onto a (fixed) plane. R (b) Find all A E M2x2(F) of rank 2 such that A2A. Similarly, find all A E M2×2(F) ofrank 0 such that A-A. (c) Find a 1-1 correspondence between the set E, := {Ae M2(F) : A2 = A and rank(A) = 1} and the set Di,I (Wo, Wi) Wo and W are linearly independent 1-dimensional subspaces of F2 1,1 :

Explanation / Answer

for part part c bases map to bases

[1 0;0 0] map to (1,0)

[[0 0;0 1] map to (0,1)

this is one- one correspondance map

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