hw1-scan - echelon or staircase-shaped forms. These echelon...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: echelon or staircase-shaped forms. These echelon forms will be studied in the next section. They will also be used for m x n systems, where m-=I- n. EXERCISES 1. Use back substitution to solve each of the following systems of equations. (a) Xl- 3X2- 2 2X2- 6 (b) Xl + X2 + X3- 8 2X2 + X3 5 3X3 9 (c) Xl + 2X2 + 2X3 + X4 5 3X2 + X3- 2X4 1-X3 + 2X4 =-1 4X4- 4 Xl + X2 + X3 + X4 + XS = 5 2X2 + X3 2X4 + XS 4X3 + X4- 2xs X4- 3xs = 2xs- 2 2. Write out the coefficient matrix for each of the systems in Exercise l. 3. In each of the following systems, interpret each equation as a line in the plane. For each system, graph the lines and determine geometrically the number of solutions. (a) Xl + X2 = 4 (b) Xl + 2X2 = 4 2-2XI 4X2 = 4 2Xl- X2 = 3 (d) Xl + X2 6 Xl- X2-Xl + 3X2 = 3 4. Write an augmented matrix for each of the systems in Exercise 3. 5. Write out the system of equations that corresponds to each of the following augmented matrices. (a) [i ~ I ~ ] (b) [ ;-2-~ I ~] 3 1 4-~ ] [!-31 2 i]-2 3 (d) 1-5 6 1 24 6-1 1 3-2 XI + X2 + X3 + 2X4 2XI + 3X2 + X3 + 3X4 X2 + X3 + X4 (b) 2x] + X2 = 8 4x]- 3X2 = 6 (d) XI + 2X2X3 2XI X2 + X3-XI + 2X2 + 3X3 (f) 3xI + 2X2 + X3 o 7 6 6 o 2-1 3 7 4X4-1 X3 + 3X3 X2 + 2X3 2xI + X2 2XI-2xI + X2 3XI and (h) 6. Solve each of the following systems. (a) XI 2X2 = 5 3xI + X2 (c) 4xI + 3X2- 4 ~XI + 4X2 3 (e) 2xI + X2 + 3X3 4XI + 3X2 + 5X3 6xI + 5X2 + 5X3 =-3 (g) I 2 2X3 =-1 3XI + 3X2 + Xl + 2X2 + ~X3- 3 2- 2: I 2 12_ I 2:XI + X2 + SX3- TO e two systems 2xI + X2 = 3 III i 4XI + 3X2 = 5 4xI + 3X2 have the same coefficient matrix but different right-hand sides. Solve both systems simultaneously by eliminating the first entry in the second row of the augmented matrix [2 1 [3-1] 4 3 5 1 and then performing back substitutions for each of the columns corresponding to the right-hand sides. 8. Solve the two systems 2xI + 5X2 + X3 2xI + 5X2 + X3 XI + 2X2 2X3 1 9 XI + 2X2 2X3 9 9 Xl + 3X2 + 4X3 = 9 XI + 3X2 + 4X3 =-2 by doing elimination on a 3 x 5 augmented matrix and then performing two back sub- stitutions. 9. Given a system of the form-mlxI + X2 = bl-m2xI + X2 = b2 where ml, m2, bI, and b2 are constants: (a) Show that the system will have a unique solution if ml =1= m2 (b) If ml = m2, show that the system will be consistent only if h = b2 (c) Give a geometric interpretation to parts (a) and (b)....
View Full Document

This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

Page1 / 8

hw1-scan - echelon or staircase-shaped forms. These echelon...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online