By Michael Artin

ISBN-10: 0130047635

ISBN-13: 9780130047632

This booklet is for the honors undergraduate or introductory graduate path. Linear algebra is tightly built-in into the textual content.

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**Extra info for Algebra**

**Example text**

If A is invertible we can multiply both sides of the equation AX = 0 by A-I to derive X = O. So the equation AX = 0 has only the trivial solution. This shows that (c) implies (d). To prove the last implication, that (d) implies (a), we take a look at square row echelon matrices M. 19) Let M be a square row echelon matrix. Either M is the identity matrix, or its bottom row is zero. 15). Suppose that (a) does not hold for a given matrix A. Then A can be reduced by row operations to a matrix A' whose bottom row is zero.

Because a product of elementary matrices is invertible, (b) implies (c). If A is invertible we can multiply both sides of the equation AX = 0 by A-I to derive X = O. So the equation AX = 0 has only the trivial solution. This shows that (c) implies (d). To prove the last implication, that (d) implies (a), we take a look at square row echelon matrices M. 19) Let M be a square row echelon matrix. Either M is the identity matrix, or its bottom row is zero. 15). Suppose that (a) does not hold for a given matrix A.

19) Corollary. 10) continue to hold if the word row is replaced by column throughout. 0 4. 2) p:S~S. Section 4 Permutation Matrices 25 is a permutation of the set {1, 2, 3}. It is called a cyclic permutation because it operates as There are several notations for permutations. We will use function notation in this section, so that p (x) denotes the value of the permutation p on the element x. 2), then p(l) = 3, p(2) = 1, p(3) = 2. A permutation matrix P is a matrix with the following property: The operation of left multiplication by P is a permutation of the rows of a matrix.

### Algebra by Michael Artin

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