Linear Algebra I & Mathematics Tutorial 1b Nagoya University, G30 Program Fall 2020 Instructor: Henrik Bachmann Homework 1: Linear systems • Deadline: 18th October, 2020 Exercise 1. (1 Point) Solve the Exercises 2 & 3 and write the solutions down by hand (paper, tablet) or by computer (Latex). Create one pdf-file (for example, by using a scanner app on your phone) and send it before the deadline ends (any time on 18th October is fine) to [email protected]. Use the following format as a filename: ”Familyname Firstname LA1 HW1.pdf”. A linear system is said to be on row-reduced echelon form if the following three conditions are satisfied: (i) The first (that it, the leftmost) variable in each equation has coefficient 1. (ii) If x i is the first variable in one of the equations, then it does not occur in any other equation in the system. (iii) If x i is the first variable in one equation, then the equations below it do not contain any of the variables x 1 ,x 2 ,...,x i-1 . Exercise 2. (2+2+2+1+1 = 8 Points) Which of the following linear systems are on row-reduced echelon form? For those that are not, find an equivalent system (i.e. one which has the same solutions) that is on row-reduced echelon form. For each system, find all solutions. i) ⇢ x 1 + x 2 + x 3 + 2x 4 = 0 x 2 - x 4 = 0 ii) 8 < : x 1 + 4x 2 + 7x 3 = 1 2x 1 + 5x 2 + 8x 3 = 2 3x 1 + 6x 2 + 10x 3 = 1 iii) x 1 +2x 2 +3x 3 +4x 4 =5 iv) 8 < : x 1 = 2 x 2 = 0 x 3 = 2 v) ⇢ x 1 + 3x 2 = 1 3x 1 + 9x 2 = 2 Exercise 3. (5 Points) Decide for which real numbers a 2 R the following linear system has solutions. Give all the solutions in these cases. 8 < : 2x 1 + 12x 2 + 7x 3 = 12a +7 2x 1 + 4x 2 + 2x 3 = 12a x 1 + 10x 2 + 6x 3 = 7a +8 . Version: October 3, 2020 -1- Solution