By S. A. Joni

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**Additional resources for Umbral Calculus and Hopf Algebras (Contemporary Mathematics, V. 6)**

**Example text**

1 Ϫ2 ΅ ΅ 1 1 1 4 2 0 3 4. ͓Ϫ1͔ 3 1 1 0 ΅ Ϫ10͔ 4 In Exercises 7–10, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Elementary Row Operations Original Matrix 7. ΄Ϫ23 5 Ϫ1 New Row-Equivalent Matrix ΅ 8. ΄ Ϫ1 3 New Row-Equivalent Matrix Ϫ4 7 ΅ ΄ Original Matrix ΄ 0 9. Ϫ1 4 Ϫ1 3 Ϫ5 ΅ 13 Original Matrix 3 Ϫ4 0 Ϫ39 Ϫ1 Ϫ8 ΄3 1 Ϫ8 Ϫ1 0 3 5 Ϫ4 Ϫ5 ΅ New Row-Equivalent Matrix Ϫ5 Ϫ7 1 5 6 3 Original Matrix ΅ ΄ Ϫ1 3 Ϫ7 0 Ϫ1 Ϫ5 0 7 Ϫ27 6 5 27 ΅ New Row-Equivalent Matrix ΄ Ϫ1 Ϫ2 3 Ϫ2 10.

Taking the natural logarithms of the given distances and periods produces the following results. 8. 8 From ln y ϭ 32 ln x, it follows that y ϭ x 3͞2, or y 2 ϭ x 3. In other words, the square of the period (in years) of each planet is equal to the cube of its mean distance (in astronomical units) from the Sun. Johannes Kepler first discovered this relationship in 1619. LINEAR ALGEBRA APPLIED Researchers in Italy studying the acoustical noise levels from vehicular traffic at a busy three-way intersection on a college campus used a system of linear equations to model the traffic flow at the intersection.

4x Ϫ 2y ϩ 5z ϭ 16 xϩ y ϭ 0 Ϫx Ϫ 3y ϩ 2z ϭ 6 Equation 1 Equation 2 Equation 3 Solve the systems provided by (a) Equations 1 and 2, (b) Equations 1 and 3, and (c) Equations 2 and 3. (d) How many solutions does each of these systems have? 52. Assume the system below has a unique solution. a11 x 1 ϩ a12 x 2 ϩ a13 x 3 ϭ b1 a21 x 1 ϩ a22 x 2 ϩ a23 x 3 ϭ b2 a31 x 1 ϩ a32 x 2 ϩ a33 x 3 ϭ b3 Equation 1 Equation 2 Equation 3 Row Equivalence In Exercises 53 and 54, find the reduced row-echelon matrix that is row-equivalent to the given matrix.