Author Question: Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all ... (Read 631 times)

DelorasTo

  • Hero Member
  • *****
  • Posts: 548
Suppose a nonhomogeneous system of 15 linear equations in 17 unknowns has a solution for all possible constants on the right side of the equation.
 
  Is it possible to find 3 nonzero solutions of the associated homogeneous system that are linearly independent? Explain.

Question 2

A mathematician has found 5 solutions to a homogeneous system of 40 equations in 42 variables. The 5 solutions are linearly independent and all other solutions can be constructed by adding together appropriate multiples of these 5 solutions.
 
  Will the system necessarily have a solution for every possible choice of constants on the right side of the equation? Explain.



nanny

  • Sr. Member
  • ****
  • Posts: 313
Answer to Question 1

No.
Since every nonhomogeneous equation Ax = b has a solution, Col A spans . So Dim Col A = 15. By the Rank Theorem, dim Nul A = 17 - 15 = 2. So the associated homogeneous system does not have more than 2 linearly independent solutions.

Answer to Question 2

No.
Let A be the 40  42 coefficient matrix of the system. The 5 solutions are linearly independent and span Nul A, so dim Nul A = 5. By the Rank Theorem, dim Col A = 42 - 5 = 37. Since 37 < 40, Col A does not span . So not every nonhomogeneous equation Ax = b has a solution.



Related Topics

Need homework help now?

Ask unlimited questions for free

Ask a Question


 

Did you know?

The most destructive flu epidemic of all times in recorded history occurred in 1918, with approximately 20 million deaths worldwide.

Did you know?

Women are two-thirds more likely than men to develop irritable bowel syndrome. This may be attributable to hormonal changes related to their menstrual cycles.

Did you know?

Recent studies have shown that the number of medication errors increases in relation to the number of orders that are verified per pharmacist, per work shift.

Did you know?

Pope Sylvester II tried to introduce Arabic numbers into Europe between the years 999 and 1003, but their use did not catch on for a few more centuries, and Roman numerals continued to be the primary number system.

Did you know?

A serious new warning has been established for pregnant women against taking ACE inhibitors during pregnancy. In the study, the risk of major birth defects in children whose mothers took ACE inhibitors during the first trimester was nearly three times higher than in children whose mothers didn't take ACE inhibitors. Physicians can prescribe alternative medications for pregnant women who have symptoms of high blood pressure.

For a complete list of videos, visit our video library