mat67-2011-midterm-solutions

mat67-2011-midterm-solutions - Solutions to Midterm Exam...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Solutions to Midterm Exam Math 67 UC Davis, Fall 2011 Problem 1 (a) Find the general form of the solution of the following system of linear equations: - x 1 + x 2 - 5 x 3 + 2 x 4 = 0 2 x 1 - x 2 + 7 x 3 - 24 x 4 = 0 x 1 + x 2 - x 3 - 2 x 4 = 0 - x 1 + 2 x 2 - 8 x 3 + 2 x 4 = 0 Solution. Write the coefficient matrix and perform Gaussian elimination to get the matrix to reduced row-echelon form: - 1 1 - 5 2 2 - 1 7 - 24 1 1 - 1 - 2 - 1 2 - 8 2 R 1 ← - R 1 R 2 R 2 - 2 R 1 R 3 R 3 - R 1 R 4 R 4 + R 1 ----------→ 1 - 1 5 - 2 0 1 - 3 - 20 0 2 - 6 0 0 1 - 3 0 R 1 R 1 + R 2 R 3 R 3 - 2 R 2 R 4 R 4 - R 2 ----------→ 1 0 2 - 22 0 1 - 3 - 20 0 0 0 40 0 0 0 20 R 3 1 40 R 3 R 4 R 4 - 20 R 3 -----------→ 1 0 2 0 0 1 - 3 0 0 0 0 1 0 0 0 0 From the RREF we see that the solution set can be written as { ( - 2 x 3 , 3 x 3 ,x 3 , 0) : x 3 R } = { x 3 ( - 2 , 3 , 1 , 0) : x 3 R } = span { ( - 2 , 3 , 1 , 0) } (b) If there is a solution except the trivial solution x 1 = x 2 = x 3 = x 4 = 0, write explicitly one other solution of the system. That is, find some specific numbers x 1 ,x 2 ,x 3 ,x 4 , not all of them zero, which solve the system. Solution. The vector ( - 2 , 3 , 1 , 0) (which is obtained from the general solution by setting x 3 = 1) is a nonzero solution. (c) The set of solutions is a subspace of
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

Page1 / 5

mat67-2011-midterm-solutions - Solutions to Midterm Exam...

This preview shows document pages 1 - 2. Sign up to view the full document.

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