This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Solutions to Assignment 1 Math 217, Fall 2002 1.1.25 Find an equation involving g , h , and k that makes this augmented matrix correspond to a consistent system: 1 4 7 g 3 5 h 2 5 9 k . To see if this matrix is consistent, we put it in row reduced form. First label the rows R 1 , R 2 , and R 3 . Then 1 4 7 g 3 5 h 2 5 9 k l R 3 R 3 + 2 R 1 1 4 7 g 3 5 h 3 5 k + 2 g l R 3 R 3 + R 2 1 4 7 g 3 5 h k + 2 g + h . We see that this matrix will be consistent if and only if k + 2 g + h = 0. 1.1.27 Suppose that the system below is consistent for all possible values of f and g . What can you say about the coefficients c and d ? Justify your answer. x 1 + 3 x 2 = f cx 1 + dx 2 = g. This system corresponds to the augmented matrix: 1 3 f c d g 1 which has row echelon form: 1 3 f d 3 c g cf . If d = 3 c , then this matrix is inconsistent whenever g cf 6 = 0 (take g = 1, f = 0, for instance). Because this matrix is supposed to be consistent for all f and g , we can conclude that d 6 = 3 c . 1.2.20 Choose h and k such that the system below has (a) no solution, (b) a unique solution, and (c) many solutions. x 1 + 3 x 2 = 2 3 x 1 + hx 2 = k. This system corresponds to the augmented matrix: 1 3 2 3 h k . which has row echelon form: A = 1 3 2 h 9 k 6 . (a) If h = 9 and k = 7, then A becomes 1 3 2 0 0 1 , which is inconsistent. Thus for h = 9 and k = 7, the system above has no solution. More generally, this system has a no solution whenever h = 9 and k 6 = 6. (b) If k = 10, then A becomes 1 3 2 0 1 k 6 , and because every column contains a pivot row, this gives a unique solution for all choices of k (for instance, k = 0 works just fine). Formally, the system given above has a unique solution for h = 10, k = 0. More generally, this system has a unique solution for all k and h such that h 6 = 9. (c) Finally, if h = 9 and k = 6, then A becomes 1 3 2 0 0 0 , which is consistent and contains a free column. Thus the system given above has a free variable, and hence, infinitely many solutions. 2 1.2.28 What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution? To know that a system has a unique solution, you need to know that it is consistent (so no pivot in the last column of the augmented matrix), and that there are no free variables (so a pivot position in each column of the coefficient matrix). Note that this also means there can not be less rows then columns (see theorem 8, pg. 69)....
View Full
Document
 Spring '08
 Neely

Click to edit the document details