# HW1 - Math E-21b Spring 2011 Homework#1 Problems due in...

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1 Math E-21b – Spring 2011 – Homework #1 Problems due in class on Thurs, Feb 3 : Section 1.1 : In exercise 11, find all solutions of the linear systems. Represent your solutions graphically, as intersections of lines in the xy -plane. 11. 22 351 7 xy      In Exercise 16, find all solutions of the linear system. Describe your solution in terms of intersecting planes. You need not sketch these planes. 16. 40 41 370 72 21 30 z z z   Note : In understanding the relationship between linear equations and the geometry of planes in R 3 , it is helpful to understand the dot product , a topic covered in Math E-21a. There’s a brief summary of vectors, the dot product and the cross product in Appendix A of the Bretscher text, but it’s best to consult any standard multivariable calculus text for a more complete treatment of these topics. A supplement is posted on the course website. We can also go over this in more detail outside of class. 17. Find all solutions of the linear system 2 35 x ya x yb , where a and b are arbitrary constants. 25. Consider the linear system 2 3 5 13 18 25 xyz z x yz k   , where k is an arbitrary number. a. For which value(s) of k does this system have one or infinitely many solutions? b. For each value of k you found in part a, how many solutions does the system have? c. Find all solutions for each value of k . 29. Find the polynomial of degree 2 [a polynomial of the form 2 () f ta b t c t ] whose graph goes through the points (1, 1) , (2,3) , and (3,13) . Sketch the graph of this polynomial. Section 1.2 : In exercise 9 and 11, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. 9. 45 6 12 5 6 123 5 6 20 2 xx x x x xxx x x . 11. 13 4 234 1234 24 8 36 3468 0 34 1 2 x xxxx       20. We say that two n m matrices in reduced row-echelon form are of the same type if they contain the same number of leading 1’s in the same positions. For example, 120 001    and 130 are of the same type. How many types of 2 2 matrices in reduced row-echelon form are there?

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HW1 - Math E-21b Spring 2011 Homework#1 Problems due in...

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