By David R. Finston and Patrick J. Morandi
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Extra resources for An Introduction to Abstract Algebra via Applications
Suppose there are elements 0 and 00 in R that satisfy a + 0 = a and a + 00 = a for all a 2 R. Recalling that addition is commutative, we have a + 0 = 0 + a = a and a + 00 = 00 + a = a for any a. If we use the …rst pair of equations for a = 00 , we get 00 + 0 = 00 . On the other hand, if we use the second pair with a = 0, we get 00 + 0 = 0. Thus, 0 = 00 . This shows that there can be only one additive identity for R. We next remark that the additive inverse of any element is unique, which is left to a homework exercise.
Suppose that e = (u; 0); with wt(u) v = w + (s; 0). 3. Show that s = u, and conclude that 2. Suppose that e = (u; ei ) with wt(u) 2. Show that s = u + bi . Conclude that wt(s) > 3 and wt(s + bi ) 2, and that v = w + (s + bi ; ei ). 3. Suppose that e = (0; u) with wt(u) 3. Show that s is the sum of at most three of the bi and that u = Bs. Conclude that wt(s) > 3 but wt(Bs) 3, and that v = w + (0; Bs). 4. Suppose that e = (ei ; u) with wt(u) 2. Show that s = ei + Bu, and that Bs = bi + u. Conclude that wt(s) > 3, wt(s + bi ) > 2 for any i, and that e = (ei ; Bs + bi ), so v = w + (ei ; Bs + bi ).
3. Let V be the set of all n m matrices with real number entries. We will denote by (aij ) the matrix whose i; j entry is aij . Recall that matrix addition and scalar multiplication are de…ned by (aij ) + (bij ) = (a ij + bij ); (aij ) = ( aij ): In other words, you add and do scalar multiplication componentwise. You should refer to a linear algebra text if necessary to recall that matrix arithmetic satis…es the properties required to see that V is an R-vector space. 4. We generalize the previous example a little.