Class Notes (838,951)
Canada (511,158)
Mathematics (182)
MATH 125 (100)
Scott (32)
Lecture 13

MATH 125 Lecture 13: Lecture 13.PDF

8 Pages
62 Views
Unlock Document

Department
Mathematics
Course
MATH 125
Professor
Scott
Semester
Fall

Description
Recap Consider a linear system of m equations in n variables with augmented matrix Alb The system is inconsistent the last non-zero row of a row echelon form of AID has the form 0 0 O c for some non-zero c E IR. If the system is consistent, it has a unique solution every variable in the system is leading, i.e every column of a row echelon form of A contains a leading entry. Otherwise, the system has infinitely many solutions. Equivalently Inconsistent rank(A) rank Alb Unique so rank(A) rank(LAlb) n Infinitely many sols rank A rank [Alb) n. Example 2.10: Consider the linear system where a and b are unknown real numbers. Determine all con ditions on a and b so that the system has (1) No solutions. (2) A unique solution. (3) Infinitely many solutions. Solution: (Warning: Don't divide by unknown numbers, or multiply a row by unknown numbers like a or b!!!) 3 2 2 rz-r 1 2 4 3 0 1 3 a b O 2 a-3 b-2 D 0 a-5 b-4 13-2m2 (1) No solution a -S o and b-4 o I ast row 0 0 0 a 5 and b 4 (2) Unique solution consistent but no free variable b can tnke an VA twe Recap Consider a linear system of m equations in n variables with augmented matrix Alb The system is inconsistent the last non-zero row of a row echelon form of AID has the form 0 0 O c for some non-zero c E IR. If the system is consistent, it has a unique solution every variable in the system is leading, i.e every column of a row echelon form of A contains a leading entry. Otherwise, the system has infinitely many solutions. Equivalently Inconsistent rank(A) rank Alb Unique so rank(A) rank(LAlb) n Infinitely many sols rank A rank [Alb) n. Example 2.10: Consider the linear system where a and b are unknown real numbers. Determine all con ditions on a and b so that the system has (1) No solutions. (2) A unique solution. (3) Infinitely many solutions. Solution: (Warning: Don't divide by unknown numbers, or multiply a row by unknown numbers like a or b!!!) 3 2 2 rz-r 1 2 4 3 0 1 3 a b O 2 a-3 b-2 D 0 a-5 b-4 13-2m2 (1) No solution a -S o and b-4 o I ast row 0 0 0 a 5 and b 4 (2) Unique solution consistent but no free variable b can tnke an VA twe(3) Infinitely many solutions consistent with at least one free variable a and Theorem 2.11: A consistent linear system which involves re variables than equations has infinitely many solutions. Proof: leading variables non -zero rows of re total rows re tion equations assump totul 4 variables Must be at least one free variable To apply this theorem, we need examples of systems which are always consistent. Definition: A system of linear equations whose constant terms are all equal to 0 is said to be homogeneous Example: The following linear system is homogeneous 4r1 3r2 Observation: A homogeneous linear system is always con sistent, because the vector is clearly a solution of any such system (we call this the trivial solution By the above dis cussion, if a homogeneous linear system has any non-trivial lutions, then it has infinitely many solutions. By Theorem 2.11, we have the following criterion for the exis- tence of infinitely many solutions: Theorem 2.12: (Solutions of homogeneous systems) A homogeneous linear system which involves more variables than equations has infinitely many solutions. Back to geometry The methods developed above allow us to address some basic geometric problems which arose in Chapter I Example 2.13: The planes P1 2y z 3 and P2 2 3y 52 1 in R3 intersect in a line l. Find a vector equation of that line l Solution: The intersechon points an Pr correspond to He Solutions row 2 is a tre variable (3) Infinitely many solutions consistent with at least one free variable a and Theorem 2.11: A consistent linear system which involves re variables than equations has infinitely many solutions. Proof: leading variables non -zero rows of re total rows re tion equations assump totul 4 variables Must be at least one free variable To apply this theorem, we need examples of systems which are always consistent. Definition: A system of linear equations whose constant terms are all equal to 0 is said to be homogeneous Example: The following linear system is homogeneous 4r1 3r2 Observation: A homogeneous linear system is always con sistent, because the ve
More Less

Related notes for MATH 125

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit