Chapter Vectors

Transcription

1 Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it meas to have a liearly idepedet set of vectors ad 5. fid the ra of a set of vectors. What is a vector? vector is a collectio of umbers i a defiite order. If it is a collectio of umbers it is called a -dimesioal vector. So the vector give by a a a is a -dimesioal colum vector with compoets a a... a. The above is a colum vector. row vector [B] is of the form B [ b b... b ] where B is a - dimesioal row vector with compoets b b... b. Example Give a example of a -dimesioal colum vector. ssume a poit i space is give by its ( x y z) coordiates. The if the value of x y z 5 the colum vector correspodig to the locatio of the poits is x y. z 5 4..

2 4.. Chapter 4. Whe are two vectors equal? Two vectors ad B are equal if they are of the same dimesio ad if their correspodig compoets are equal. Give a a ad the a b b B b B if a i b i.... i Example What are the values of the uow compoets i B if 4 ad b B 4 b4 ad B. b b 4 How do you add two vectors? Two vectors ca be added oly if they are of the same dimesio ad the additio is give by

3 Vectors 4.. b b b a a a B ] [ ] [ a b b a b a Example dd the two vectors 4 ad 7 5 B B

4 4..4 Chapter 4. Example 4 store sells three brads of tires: Tirestoe Michiga ad Copper. I quarter the sales are give by the colum vector where the rows represet the three brads of tires sold Tirestoe Michiga ad Copper respectively. I quarter the sales are give by 6 What is the total sale of each brad of tire i the first half of the year? The total sales would be give by C So the umber of Tirestoe tires sold is 45 Michiga is 5 ad Copper is i the first half of the year. What is a ull vector? ull vector (also called zero vector) is where all the compoets of the vector are zero. Example 5 Give a example of a ull vector or zero vector. The vector

5 Vectors 4..5 is a example of a zero or ull vector. What is a uit vector? uit vector U is defied as u u u U where u u u u Example 6 Give examples of -dimesioal uit colum vectors. Examples iclude etc. How do you multiply a vector by a scalar? If is a scalar ad is a -dimesioal vector the a a a a a a

6 4..6 Chapter 4. Example 7 What is if Example 8 store sells three brads of tires: Tirestoe Michiga ad Copper. I quarter the sales are give by the colum vector If the goal is to icrease the sales of all tires by at least 5% i the ext quarter how may of each brad should be sold? Sice the goal is to icrease the sales by 5% oe would multiply the vector by.5 5 B Sice the umber of tires must be a iteger we ca say that the goal of sales is

7 Vectors B What do you mea by a liear combiatio of vectors? Give m... as m vectors of same dimesio ad if m... are scalars the m m... is a liear combiatio of the m vectors. Example 9 Fid the liear combiatios a) B ad b) C B where 6 C B a) 6 B 6 4 b) 6 C B 6 6

8 4..8 Chapter 4. 7 What do you mea by vectors beig liearly idepedet? set of vectors m are cosidered to be liearly idepedet if... m m has oly oe solutio of... m Example re the three vectors liearly idepedet? Writig the liear combiatio of the three vectors gives The above equatios have oly oe solutio. However how do we show that this is the oly solutio? This is show below. The above equatios are 5 5 () 64 8 () 44 () Subtractig Eq () from Eq () gives 9 (4) Multiplyig Eq () by 8 ad subtractig it from Eq () that is first multiplied by 5 gives

9 Vectors (5) Remember we foud Eq (4) ad Eq (5) just from Eqs () ad (). Substitutio of Eqs (4) ad (5) i Eq () for ad gives 44 ( ) 4 8 This meas that has to be zero ad coupled with (4) ad (5) ad are also zero. So the oly solutio is. The three vectors hece are liearly idepedet. Example re the three vectors liearly idepedet? By ispectio or So the liear combiatio has a o-zero solutio Hece the set of vectors is liearly depedet. What if I caot prove by ispectio what do I do? Put the liear combiatio of three vectors equal to the zero vector to give 6 () 5 4 () () Multiplyig Eq () by ad subtractig from Eq () gives

10 4.. Chapter 4. (4) Multiplyig Eq () by.5 ad subtractig from Eq () gives.5 (5) Remember we foud Eq (4) ad Eq (5) just from Eqs () ad (). Substitute Eq (4) ad (5) i Eq () for ad gives 5( ) 7( ) This meas ay values satisfyig Eqs (4) ad (5) will satisfy Eqs () () ad () simultaeously. For example chose 6 the from Eq (4) ad from Eq (5). Hece we have a otrivial solutio of [ ] [ 6]. This implies the three give vectors are liearly depedet. Ca you fid aother otrivial solutio? What about the followig three vectors? re they liearly depedet or liearly idepedet? Note that the oly differece betwee this set of vectors ad the previous oe is the third etry i the third vector. Hece equatios (4) ad (5) are still valid. What coclusio do you draw whe you plug i equatios (4) ad (5) i the third equatio: 5 7 5? What has chaged? Example re the three vectors liearly idepedet? Writig the liear combiatio of the three vectors ad equatig to zero vector

11 Vectors gives I additio to oe ca fid other solutios for which are ot equal to zero. For example 4 is also a solutio as Hece are liearly depedet. What do you mea by the ra of a set of vectors? From a set of -dimesioal vectors the maximum umber of liearly idepedet vectors i the set is called the ra of the set of vectors. Note that the ra of the vectors ca ever be greater tha the vectors dimesio. Example What is the ra of ? 44 Sice we foud i Example. that are liearly idepedet the ra of the set of vectors is. If we were give aother vector 4 the ra of the set of the vectors 4 would still be as the ra of a set of vectors is always less tha or equal to the dimesio of the vectors ad that at least are liearly idepedet. Example 4 What is the ra of ?

12 4.. Chapter 4. I Example. we foud that are liearly depedet the ra of is hece ot ad is less tha. Is it? Let us choose two of the three vectors Liear combiatio of ad equal to zero has oly oe solutio the trivial solutio. Therefore the ra is. Example 5 What is the ra of? 4 5 From ispectio that implies. Hece. has a otrivial solutio. So are liearly depedet ad hece the ra of the three vectors is ot. Sice ad are liearly depedet but. has trivial solutio as the oly solutio. So ad are liearly idepedet. The ra of the above three vectors is. Prove that if a set of vectors cotais the ull vector the set of vectors is liearly depedet. Let... m be a set of -dimesioal vectors the m m is a liear combiatio of the m vectors. The assumig if is the zero or ull vector ay value of coupled with will satisfy the above equatio. Hece the m

13 Vectors 4.. set of vectors is liearly depedet as more tha oe solutio exists. Prove that if a set of m vectors is liearly idepedet the a subset of the m vectors also has to be liearly idepedet. Let this subset of vectors be a a ap where p < m. The if this subset of vectors is liearly depedet the liear combiatio a a p ap has a o-trivial solutio. So a a p ap a ( p )... am also has a o-trivial solutio too where a ( p ) am are the rest of the ( m p) vectors. However this is a cotradictio. Therefore a subset of liearly idepedet vectors caot be liearly depedet. Prove that if a set of vectors is liearly depedet the at least oe vector ca be writte as a liear combiatio of others. Let m be liearly depedet set of vectors the there exists a set of scalars m ot all of which are zero for the liear combiatio equatio m m. Let p be oe of the o-zero values of i i m that is p the p p m p p p m. p p p p ad that proves the theorem. Prove that if the dimesio of a set of vectors is less tha the umber of vectors i the set the the set of vectors is liearly depedet. Ca you prove it? How ca vectors be used to write simultaeous liear equatios? If a set of m simultaeous liear equatios with uows is writte as ax a x c ax a x c a x a x c m m

14 4..4 Chapter 4. where where where x x x x x are the uows the i the vector otatio they ca be writte as a a m a a m a a m a a m c C c m x C The problem ow becomes whether you ca fid the scalars combiatio x... x is equal to the C that is x... x C x x... x such that the liear Example 6 Write 5x 5x x x 8x x x x x 79. as a liear combiatio of set of vectors equal to aother vector.

15 Vectors x 5x x x 8x x x 79. x x x 64 x 8 x What is the defiitio of the dot product of two vectors? a a B b b be two -dimesioal vectors. The the dot Let [ ] ad [ ] a b product of the two vectors ad B is defied as B a b a b a b a b i dot product is also called a ier product. Example 7 Fid the dot product of the two vectors [4 ] ad B [ 7 ]. B [4].[7] (4)()()()()(7)()() Example 8 product lie eeds three types of rubber as give i the table below. Rubber Type Weight (lbs) Cost per poud ($) B C Use the defiitio of a dot product to fid the total price of the rubber eeded. The weight vector is give by W [5] ad the cost vector is give by C [..569.]. The total cost of the rubber would be the dot product of W ad C. W C [ 5] [..569.] i i

16 4..6 Chapter 4. ( )(.) (5)(.56) ()(9.) $7. Key Terms: Vector dditio of vectors Ra Dot Product Subtractio of vectors Uit vector Scalar multiplicatio of vectors Null vector Liear combiatio of vectors Liearly idepedet vectors