By Michael Artin

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

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**Extra resources for Algebra**

**Example text**

If Xi = xi with 0 <: i < j < m, then x i - i = 1. ) Moreover, any power x n is equal to one of them: By division with remainder, we may write n = mq + r with remainder r less than m. Then x n = (xm)qx r = x r . Thus H consists of the following m elements: H = {1, x, ... 9) 1 }, these powers are distinct, and x m = 1. Such a group is called a cyclic group of order m. The order of any group G is the number of its elements. 10) I G 1 = number of elements of G. Of course, the order may be infinite.

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.

Find all solutions of the equation Xr + X2 + 2X3 - X4 :=: 3. 4. 22) and verify that their product is A -1. 5. Find inverses of the following matrices: 6. Make a sketch showing the effect of multiplication by the matrix A = [; 7. 8. 9. 10. 11. 12. 13. ~] on the plane ~2. How much can a matrix be simplified if both row and column operations are allowed? (a) Compute the matrix product eijeu. (b) Write the identity matrix as a sum of matrix units. (e) Let A be any n X n matrix. Compute eiiAejj. (d) Compute eijA and Aeij.