Class Notes (835,006)
Canada (508,865)
Mathematics (1,919)
MATH 136 (168)
Lecture

lect136_2_w14.pdf

5 Pages
96 Views
Unlock Document

Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
Wednesday, January 8 − Lecture 2 : Linear independence of vectors in ℝ . n Concepts: 1. Linearlyindependent subset in ℝ . n 2. Characterization of a linearly independent set as one being a set where no vector is a linear combination of the others. 3. A plane in ℝ . A hyperplane in ℝ . n 2.1 Definitions. - Let v 1, v2,.., v ke k vectors in ℝ and suppose the vector 0 represents the zero vector 0 = (0, 0, …, 0). The vectors v , v 1...2,v areksaid to be linearly independentif the only way that can hold true is if α ,1α ,2..., α ake all zeroes. The solution where all the α‘s are i zeros is called the trivial solution of this vector equation. - If v , v ..., v are not linearly independent then they are said to be linearly 1 2, k dependent. Note that a linearly independent set cannot contain the zero-vector, 0. This fact follows from the definition. In class, we will often abbreviate the words “linearly independent” with the letters “L.I.”. 3 2.2 Example − Verify whether the set {v 1, v2, v3} in ℝ where v = (1, 1, 1), v = (02 1, 7) and v 3 (0, 0, 3) is linearly independent. Solution: The given set is linearly independent. m 2.3 Theorem − A subset of a finite linearly independent subset of ℝ is linearly independent. Proof: m - Suppose M = {v , v , 1..,2v } is r linearly independentsubset of ℝ . - Let S = {v , k1, .k2., v } km a non-empty subset of M. We claim that S is linearly independent. - Suppose not. Then there exists α , α , k1 k2 …, α km ,not all equal to zero, such that α k1+k1 v + …k2 k2v = 0. km km - Suppose, without loss of generality, α ≠ 0. Then k1 0v 1 … + (α v + αk1 k1… + αk2 k2+ ... + 0v =km.km r where α ≠ k1 - This contradicts the fact that M = {v , v , ..1., 2 } is a rinearly independent subset of m ℝ . - The source of the contradiction is our supposition that S is not linearly independent. - So S = {v k1, vk2 ...., vkm is linearly independent, as required. 2.4 Theorem. An important characterization of linear independence − The vectors M = {v , v , ...., v } of ℝ n with k ≥ 2, are linearly independent ifand only if no vector 1 2 k , in M is a linear combination of the others. Proof of (⇐): Given: None of these vectors in M is a linear combination of the others. n Required to show: That M is linearly independent subset of ℝ . Let’s suppose that M is not linearly independent. - Then there exists α , α 1 ...2 α not akl zeroes such that - Suppose one of the coefficients, sayα is not zerop,Rearrange the order of {v , v , ...., 1 2 vk} so that p = 1, .i.e., α is 1ot zero. - Then - Hence v is1a linear combination of the others. Contradiction. - So M is linearly independent. Proof of (⇒) : Given: M = {v 1, v2, ...., k } is linearly independent. Required to show: That no vector in M is a linear combination of the others. - Suppose one vector of {v , v , 1...2 v } is k linear combination of the others. Say it is v1. We claim that this will lead to a contradiction of our hypothesis. - Then there exists α ...2., α sukh that such that - Then - This contradicts the fact that M is linearly independent. - Then no vector of {v , v ,1....2 v } iska linear combination of the others. 2.5 Remark − The above theorem shows that we may have defined “linearly independent” as follows: n "The set U = {v , v1...2,v } iska linearly independent of ℝ if and only if no vector in U is a linear combination of the other vectors in U." If a vector v jin U is a linear combination of the others we refer to v as
More Less

Related notes for MATH 136

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