MATH 4130 Midterm: Exam 1
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You may use any theorems which were proved in class or in the textbook provided that you state them clearly. You may not use results which were proved as part of your homework unless you prove them again here: problem 1. Verify that dim(v + w ) + dim(v w ) = dim(v ) + dim(w ): problem 2. T n 1(v)} forms a basis for v . (b) find a basis for the null-space n (t ) and the range r(t ) of t and verify the dimension theorem. (c) calculate [t ] : problem 4. T 2 = t . (c) conversely, show that any linear transformation t : v v satisfying t 2 = t is a projection on r(t ) along n (t ): problem 5 c. T (a) = p a, for a m2 2(r).
