Question: ker=kernel, ran=rank, dim=dimension, null=null space. Let V be a real vector space and suppose T : V > V is a linear map with the

ker=kernel, ran=rank, dim=dimension, null=null space.

Let V be a real vector space and suppose T : V > V is a linear map with the property that T2 = T. (a) (10 points) Show that V = ker(T) $ran(T). (b) (10 points) Suppose dim(V) = 7 and null(T) = 2. Show that T has at least one non-zero eigenvalue. Hint for (b): A polynomial with real coefcients of odd degree has at least one real root

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