EE221A Problem Set 1 Solutions - Fall 2011 Note: these solutions are somewhat more terse than what we expect you to turn in, though the important thing is that you communicate the main idea of the solution. Problem 1. Functions. It is a function; matrix multiplication is well deﬁned. Not injective; easy to ﬁnd a counterexample where f ( x 1 ) = f ( x 2 ) ; x 1 = x 2 . Not surjective; suppose x = ( x 1 ,x 2 ,x 3 ) T . Then f ( x ) = ( x 1 + x 3 ,0 ,x 2 + x 3 ) T ; the range of f is not the whole codomain. Problem 2. Fields. a) Suppose00 and0 are both additive identities. Then x + 00 = x + 0 = 0 ⇐⇒00 = 0 . Suppose 1 and 10 are both multiplicative identities. Consider for x 6 = 0 , x · 1 = x = x · 10 . Premultiply by x-1 to see that 1 = 10 . b) We are not given what the operations + and · are but we can assume at least that + is componentwise addition. The identity matrix I is nonsingular so I ∈ GL n . But I + (-I ) = 0 is singular so it cannot be a ﬁeld. Problem 3. Vector Spaces.
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