Algebra by Michael Artin

By Michael Artin

This e-book is for the honors undergraduate or introductory graduate direction. Linear algebra is tightly built-in into the textual content.

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Prove that A -I has integer entries if and only if det A = ± l. 4. Prove that expansion by minors on a row of a matrix defines the determinant function. MisceDaneous Problems 1. Write the matrix [~ ~] as a product of elementary matrices, using as few as you can. Prove that your expression is as short as possible. 2. Find a representation of the complex numbers by real 2 x 2 matrices which is compatible with addition and multiplication. Begin by finding a nice solution to the matrix equation A 2 = -I.

1) mentions all parts of the definition of a group except for the associative law. We do not need to mention associativity. It carries over automatically from G to H. Section 2 Subgroups 45 Every group has two obvious subgroups: the whole group and the subgroup {I} consisting of the identity element alone, A subgroup is said to be a proper subgroup if it is not one of these two. 2) Examples. (a) The set T of invertible upper triangular 2 x 2 matrices (a,d*O) is a subgroup of the general linear group GLilR).

However, this is not a very important point, so we omit the proof. The reason that row reduction is useful is that we can solve a system of equations A'X = B' immediately if A' is in row echelon form. For example, suppose that [A'l B' ] = [ 1 6 0 1 0 0 1 2 o 0 0 0 There is no solution to A'X = B' because the third equation is 0 = 1. On the other hand, [A'l B ' ] = [ 1 6 0 1 0 0 1 2 000 0 ~] has solutions. Choosing X2, X4 arbitrarily, we can solve the first equation for the second for X3. 13). 16) Proposition.

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