[線代] 線代(1)

[PttFund (批踢踢基金只進不出)]

Let V be a subspace of R^n.  A linear transformation
μ: R^n -> R^n is called a projection of R^n on V if
μ(x) in V and x-μ(x) in V^⊥ for every x in R^n.


(a) Let A be a n╳n matrix.  Show that μ: R^n -> R^n
    defined by μ(x) = Ax for all x in R^n is a projection
    of R^n onto C(A), if and only if A^2 = A and A = A^T.

    (C(A) = the column space of A.)

(b) What are the eigenvalues and the corresponding
    eigenspaces of a projection?


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