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# Class Note for MATH 215 with Professor Dostert at UA 2

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Date Created: 02/06/15
THE UNIVERSITY a OF ARIZONA Math 215 Introduction to Linear Algebra Section 23 Spanning Sets and Linear Independence Paul Dostert August 18 2008 A19 Spanning Sets of Vectors Solving Ax b is closely related to the problem of determining when one vector can be written as a linear combination of others 1 2 Ex Can 3 be represented as a linear combination of 1 and 4 0 1 0 1 What about 3 1 2 Thm A system of linear equations with augmented matrix Ab is consistent iff b is a linear combination of columns of A This theorem is clear for an n gtlt n matrix A Write the columns of A as an then Ax b may then be written as 31361 anxn b 817 Ah Spanning Set 7vk is a set of vectors in Rquot then the set of all linear 7vk is called the span of v17 v27 7vk and is 7vk or span S If span S Rquot then S is called If S V17V27 combinations of v17 v27 denoted by span v17vQ7 a spanning set of R Ex ShowthatR2span 31 Ex Show that R span e17 7 en 1 1 Ex Find the span of 3 7 2 0 1 Comment In the previous example we have 2 vectors in R3 The maximum space 2 vectors can span is of R2 How can we find another vector so that the span of three vectors will be R3 A Linear Dependence Recall that if we write y fac then we call c an independent variable and y a dependent variable This is because we can write y in terms of only c variables Using this idea we can discuss the idea of linear dependence 0 In a previous example we showed that u 3 is a linear combination 2 2 1 of w 1 and v 1 by writing u w 2v We may say that u 0 1 depends on w and v This means u w and v are linearly dependent since we can always write one of the vectors as a linear combination of the others A set of vectors v1 vk is linearly dependent if there are scalars cl ck at least one of which is not zero such that Clvl Ckvk 0 A set of vectors that is not linearly dependent is called linearly independent Why do we have the phrase at least one of which is not zero in the previous definition A Linear Dependence The following theorem shows the definition of linear dependence coincides with the idea given for its motivation Thm Vectors v1 vm in R are linearly dependent iff at least one of the vectors can be expressed as a linear combination of the others Ex Give a simple proof of this theorem using the definition of linearly dependence as given in the previous slide Ex Determine whether the following are linearly independent and and and UlODl ll lOl lUl llgt l ll l I l A 039 v l l WNl Ol l Ol l l l l l l IUlNl l ONODll Ah Showing Linear Independence To show a set of vectors are linearly independent we use the following theorem Thm Let v17 7vm be column vectors in R and let A be the n gtlt m matrix v1 vm with these vectors as its columns Then v17 7vm are linearly dependent iff AO or Ax 0 has a nontrivial solution x 7 0 If n 7 m there are some theorems we can use to easily determine if vectors are linearly dependent Thm Let v17 V1 7vm be row vectors in R and let A be the m gtlt n matrix with these vectors as its rows Then v17 7vm are linearly Vm dependent iff rank A lt m Thm Any set of m vectors in R is linearly dependent if m gt n A19 Examples for Showing Linear Dependence 0 2 5 using the previous theorems 1 1 4 Ex Show the vectors 1 7 3 and 9 are linearly dependent by Ex Show the vectors 1 7 g the previous theorems first then show they are linearly dependent by solving the homogeneous linear system and 2 are linearly dependent Use Ex Show that if u and v are linearly independent vectors then so are u v and u v A Matlab Examples 1 1 4 Once again we will show 1 3 and 9 0 2 5 are linearly independent Matlab has a rank function which estimates the rank of a matrix For small matrices this will always give the exact rank For large matrices with bad values some very small and very large values this will not always work correctly The construct is quite easy as the rank function simply takes a matrix as an argument gtgt A 110132495 rankA Using the theorem if rank A lt 3 then the vectors are linearly independent The other option is to use the rref which reduces A to reduced row echelon form A 110132495 rrefA If the rref of A has any zero rows then the set of vectors are linearly dependnent

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