hw01 - Fall 2012 Math 419 Problem Set 1 Due on Thu Sep 13 1 Solve the following systems using Gauss-Jordan elimination(a 3x 4y z = 8 6x 8y 2z = 3 x1

hw01 - Fall 2012 Math 419 Problem Set 1 Due on Thu Sep 13 1...

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Unformatted text preview: Fall 2012 Math 419 Problem Set 1 Due on Thu, Sep 13 1) Solve the following systems using Gauss-Jordan elimination. (a) 3x + 4y − z = 8 , 6x + 8y − 2z = 3 x1 − 7x2 (b) (c) x3 + x5 = 3 − 2x5 = 2 , x4 + x5 = 1 x1 + 2x2 − 2x3 + x4 + 3x5 = 2 2x1 + x2 + 2x3 + x5 = 3 . − 2x1 − 3x2 + 2x3 − x4 + 2x5 = 1 2) Solve the following system by noticing that the values of x3 can be freely chosen. 3x2 + 3x3 − 2x4 = − 4 x1 + x2 + 2x3 + 3x4 = 2 x1 + 2x2 + 3x3 + 2x4 = 1 x1 + 3x2 + 4x3 + 2x4 = − 1 3) Determine which of the matrices below are in reduced row-echelon form: 12020 1203 01203 0 0 1 3 0 (a) 0 0 1 4 0 , (b) 0 0 0 1 4 , (c) 0 0 0 0 , 0012 00000 00001 (d) 0 1 2 3 4 . 123 147 4) Find the rank of the matrices. (a) 0 1 2 , (b) 2 5 8 . 001 369 5) Consider a linear system of three equations with three unknowns. We are told that the system has a unique solution. What does the reduced row-echelon form of the coefficient matrix of this system look like? Explain your answer. 1 2 5 3 and y = 0 . 6) Let x = 1 −9 (a) Find a matrix A of rank 1 such that Ax = y . (b) Find a matrix A with all nonzero entries such that Ax = y . 7) Consider a solution x1 of the linear system Ax1 = b. Justify (a) and (b): (a) If xh is a solution of the system Ax = 0, then x1 + xh is a solution of the system Ax = b. (b) If x2 is another solution of the system Ax = b, then x2 − x1 is a solution of the system Ax = 0. If the problem is difficult, you can think of the following examples: 1 3 12 . , x1 = ,b= A= 1 9 36 8) Suppose A is a 2 × 2 matrix. Let x1 be a solution vector of the system Ax = b. We are told that the solutions of the system Ax = 0 form a line which is not parallel to x1 . Draw the line consisting of all solutions of the system Ax = b. 9) Consider two vectors v1 and v2 in R3 that are not parallel. Which vectors in R3 are linear combinations of v1 and v2 ? Describe the set of these vectors geometrically. Include a sketch in your answer. 10) Let A be the n × n matrix with 0’s on the main diagonal, and 1’s everywhere else. For an arbitrary vector b in Rn , solve the linear system Ax = b, expressing the components x1 , . . . , xn of x in terms of the components of b. If n = 3, the linear system is written as y + z = b1 x + z = b2 . x+y = b3 2 ...
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