VECTORS [PARTS OF 1.3] 5-1

Transcription

1 VECTORS [PARTS OF.3] 5-

2 Vectors and the set R n A vector of dimension n is an ordered list of n numbers Example: v = [ ] 2 0 ; w = ; z = v is in R 3, w is in R 2 and z is in R? 0. 4 In R 3 the R stands for the set of real numbers that appear as entries in the vector, and the exponents 3, indicate that each vector contains 3 entries. A vector can be viewed just as a matrix of dimension m 5-2 Text:.3 Vectors 5-2

3 R n is the set of all vectors of dimension n. We will see later that this is a vector space, i.e., a set that has some special properties with respect to operations on vectors. Two vectors in R n are equal when their corresponding entries are all equal. Given two vectors u and v in R n, their sum is the vector u + v obtained by adding corresponding entries of u and v Given a vector u and a real number α, the scalar multiple of u by α is the vector αu obtained by multiplying each entry in u by α (!) Note: the two vectors must be both in R n, i.e., then both have n components. Let us look at this in detail 5-3 Text:.3 Vectors 5-3

4 Sum of two vectors x = x ; x 2 x 3 y = y y 2 ; x + y = x + y y 2 + x 2 y 3 x 3 + y 3 with numbers: x = 2 ; 3 y = 0 3 ; x + y = 5 3?? 5-4 Text:.3 Vectors 5-4

5 Multiplication by a scalar Given: a number α (a scalar ) and a vector x: α R, x R 3, αx = αx αx 2 αx 3 with numbers: α = 4; x = 2 αx = In the text vectors are represented by bold characters and scalars by light characters. We will often use Greek letters for scalars and regular latin symbols for vectors 5-5 Text:.3 Vectors 5-5

6 Properties of + and α The vector whose entries are all zero is called the zero vector and is denoted by 0. (a) x + y = y + x (Addition is commutative) (b) x + (y + z) = (x + y) + z (Addition is associative) (c) 0 + x = x + 0 = x, (0 is the vector of all zeros) (d) x + ( x) = x + x = 0 ( x is the vector ( )x) (e) α(x + y) = αx + αy (f) (α + β)x = αx + βx (g) (αβ)x = α(βx) (h) x = x for any x 5-6 Text:.3 Vectors 5-6

7 Linear combinations Very important concept.. A linear combination of m vectors is a vector of the form: x = α x + α 2 x α m x m where α, α 2,, α m, are scalars and x, x 2,, x m, are vectors in R n. The scalars α, α 2,, α m are called the weights of the linear combination They can be any real numbers, including zero 5-7 Text:.3 Vectors 5-7

8 Linear combinations Example: Linear combinations of vectors in R 3 : u = ; w = And we have: u = 2 0 ; w =?? 4? Note: for w the second weight is and the third is Text:.3 Vectors 5-8

9 The linear span of a set of vectors Definition: If v,, v p are in R n, then the set of all linear combinations of v,, v p is denoted by span{v,, v p } and is called the subset of R n spanned (or generated) by v,, v p. That is, span{v,, v p } is the collection of all vectors that can be written in the form α v + α 2 v α p v p with α, α 2,, α p scalars. [ ] 2 What is span{u} in R 2 where u =? 0 [ ] What is span{v} in R 2 where v =? What is span{u, v} in R 2 with u, v given above? 5-9 Text:.3 Vectors 5-9

10 Does the vector [ ] Same question for the vector What is span{u, v} in R 3 when: u = ; v = 0 2? 0 belong to this span{u, v}? [ ] Is span{u, v} the same as span{v, u}? Do the vectors: a = ; b = belong to span{u, v} found in the previous question.? Is span{u, v} the same as span{2u, 3v}? 5-0 Text:.3 Vectors 5-0

11 Geometric representation of R 2 and R 3 Consider a rectangular coordinate system in the plane. The illustration shows the vector [ a x = b] b x 2 x (a,b) a [ ] b with a = 4, b = 2. Each point in the plane is determined by an ordered pair of numbers, [ ] so we identify a geometric point (a, b) with the column a vector b We may regard R 2 as the set of all points in the plane a 5- Text:.3 Vectors 5-

12 R 2 x 2 (2,) x (, ) x in the horizontal direction, x 2 in vertical direction 5-2 Text:.3 Vectors 5-2

13 Often we draw an oriented line from origin to the point: (2,) (, ) 5-3 Text:.3 Vectors 5-3

14 R 3 x 3 (2,3,3) x 2 x horizontal = x 2, vertical=x 3, back to front direction = x (However some representations may differ). We will use this one. 5-4 Text:.3 Vectors 5-4

15 Geometric interpretation of addition of 2 vectors First viewpoint: Think of moving ( rigidly ) one of the vectors so its origin is at endpoint of the other vector. Then x + y is the vector from origin to the end point of the shifted vector. y x x+y x y with origin shifted 5-5 Text:.3 Vectors 5-5

16 Second viewpoint: x + y correponds to the fourth vertex of the parallelogram whose other three vertices are: O, x, and y x+y y x y x O Using the first viewpoint, show geometrically how to add the 3 vectors [ ], [ ] 2 0, and [ ] Text:.3 Vectors 5-6

17 Geometric interpretation of span{v} Let v be a nonzero vector inr 3 x 3 Then span{v} is the set of all scalar multiples of v This is also the set of points on the line in R 3 through v and 0. Span { v } v x 2 x 5-7 Text:.3 Vectors 5-7

18 Geometric interpretation of span{u, v} Let u, v be two nonzero vectors in R 3 with v not a multiple of u. Then span{u, v} is the plane in R 3 that contains u, v, and 0. In particular, span{u, v} contains the two lines span{u} and span{v} v u 2u+3v (See also Figure. from text). 5-8 Text:.3 Vectors 5-8

19 LINEAR INDEPENDENCE [.7] 5-9

20 Linear independence [Important] Definition The set {v,..., v p } is said to be linearly dependent if there exist weights c,..., c p, not all zero, such that c v + c 2 v c p v p = 0 It is linearly independent otherwise The above equation is called linear dependence relation among the vectors v,, v p Another way to express dependence: A set of vectors is linearly dependent if and only if there is one vector among them which is a linear combination of all the others. Prove this 5-20 Text:.7 LinearInd 5-20

21 Q: Why do we care about linear independence? A: When expressing a vector x as a linear combination of a system {v,, v p } that is linearly dependent, then we can find a smaller system in which we can express x A dependent system is redundant [ ] Let v =. Is {v } linearly independent? [here: p = ] A system consisting of a nonzero vector [at least one nonzero entry] is always linearly independent: True - False? {[ 0 Are the following systems linearly independent: ] [ ]} {[ ] [ ]} {[ ] 0,,,,, [ ], [ ]} 2? 5-2 Text:.7 LinearInd 5-2

22 A system {u, v} is linearly dependent when? Let v = ; v2 = 4 ; v3 = 2 3 ; 2 5 (a) Determine if {v, v 2, v 3 } is linearly independent (b) If possible find a linear dependence relation among v, v 2, v 3. Solution: We must determine if the system: x + x2 4 + x3 2 3 = has a nontrivial solution (Trivial solution: x = x 2 = x 3 = 0) 5-22 Text:.7 LinearInd 5-22

23 Augmented syst: Echelon st step This system is equivalent to original one. Variable x 3 is free. Echelon 2nd step Select x 3 = 3 (to avoid fractions) and back-solve for x 2 (x 2 = 5), and x, (x = 4) Conclusion: there is a nontrivial solution NOT independent (b) Linear dependence relation: From above, 4v + 5v 2 + v 3 = Text:.7 LinearInd 5-23

24 Note: Text uses the reduced echelon form instead of back-solving [Result is clearly the same. Both solutions are OK] With the reduced row echelon form 0 4/ / x = (4/3)x 3 ; x 2 = (5/3)x 3 select x 3 = 3 then x 2 = 5, x = 4 Recall: x, x 2 are basic variables, and x 3 is free 5-24 Text:.7 LinearInd 5-24