Announcements Wednesday, September 06. WeBWorK due today at 11:59pm. The quiz on Friday covers through Section 1.2 (last weeks material)

Transcription

1 Announcements Wednesday, September 06 WeBWorK due today at 11:59pm. The quiz on Friday coers through Section 1.2 (last eeks material)

2 Announcements Wednesday, September 06 Good references about applications(introductions to chapters in book) Aircraft design, Spacecraft controls (Ch. 2, 4) Imaging distorsion, Image processing, Computer graphics (Ch. 3,7,8) Management, Economics, Making sense of a lot of data (Ch. 1, 6) Ecology and sustainability (Ch. 5) Thermodynamics, heat transfer (Worksheet eek 1) A reference to Surely you re joking Mr. Feynman (Ch. 3) I ll try to find something for you guys: Mechanical systems, Solar panels, origami, sarm behaiour Neuroscience, Prehealth, Population groth Computer logic Optimization

3 Section 1.3 Vector Equations

4 Motiation Linear algebra s to iepoints: Algebra: systems of equations and their solution sets Geometry: intersections of points, lines, planes, etc. x 3y = 3 2x + y = 8 The geometry ill gie us better insight into the properties of systems of equations and their solution sets.

5 Vectors Elements of R n can be considered points... or ectors: arros ith a gien length and direction. the point (1, 3) the ector ( 1) 3 x-coordinate: idth of ector horizontally, y-coordinate: height of ector ertically. It is conenient to express ectors in R n as matrices ith n ros and one column: 1 = 2 3 Note: Some authors use bold typography for ectors:.

6 Vector Algebra (applies to ectors in R n ) Definition We can add to ectors together: a x a + x b + y = b + y. c z c + z We can multiply, or scale, a ector by a real number: x c x c y = c y. z c z Distinguish a ector from a real number: call c a scalar. c is called a scalar multiple of. For instance, = 7 and 2 2 =

7 Addition: The parallelogram la 5 = = Geometrically, the sum of to ectors, is obtained by creating a parallelogram: 1. Place the tail of at the head of. 2. Sum ector + has tail: tail of 3. Sum ector + has head: head of The idth of + is the sum of the idths, and likeise ith the heights. For example, ( ) ( ) ( ) = Note: addition is commutatie.

8 Geometry of ector substraction If you add to, you get. Geometrically, the difference of to ectors, is obtained as follos: 1. Place the tails of and at the same point. 2. Difference ector has tail: head of 3. Difference ector has head: head of For example, ( ) 1 4 This orks in higher dimensions too! ( ) 4 = 2 ( ) 3. 2

9 Toards linear spaces Scalar multiples of a ector: hae the same direction but a different length. The scalar multiples of form a line. Some multiples of ( ) 1 = 2 ( ) 2 2 = 4 1 ( ) 1 2 = 2 1 ( ) 0 0 = 0 All multiples of.

10 Linear Combinations We can generate ne ectors ith addition and scalar multiplication: Definition = c c c p p We call a linear combination of the ectors 1, 2,..., p, and the scalars c 1, c 2,..., c p are called the eights or coefficients. c 1, c 2,..., c p are scalars, 1, 2,..., p are ectors in R n, and so is. Example ( ( 1 1 Let = and =. 2) 0) What are some linear combinations of and?

11 Poll Poll Is there any ector in R 2 combination of and? that is not a linear No: in fact, eery ector in R 2 is a combination of and. (The purple lines are to help measure ho much of and you need to reach a gien point.)

12 Poll Poll Which of the folloing are possible shapes for the Span { 1, 2} of 2 ectors in R 3? Select all possible shapes! A Empty B Point C Line D Circle E the grid points on a 2-plane F the 4-plane Anser: B and C. (Span is neer empty, more details on Friday. and to ectors may span a 2-plane, but not only its grid points)

13 More Examples ( 2 What are some linear combinations of =? 1) What are all linear combinations of? All ectors c for c a real number. I.e., all scalar multiples of. These form a line. Question What are all linear combinations of ( ( ) 2 1 = and =? 2) 1 Anser: The line hich contains both ectors. What s different about this example and the one on the poll?

14 Span It ill be important to handle all linear combinations of a set of ectors. Definition Let 1, 2,..., p be ectors in R n. The span of 1, 2,..., p is the collection of all linear combinations of 1, 2,..., p, and is denoted Span{ 1, 2,..., p}. In symbols: Span{ 1, 2,..., p} = { x x x p p x1, x 2,..., x p in R }. In other ords: Span{ 1, 2,..., p} is the subset spanned by or generated by 1, 2,..., p. it s exactly the collection of all b in R n such that the ector equation (unknons x 1, x 2,..., x p) is consistent i.e., has a solution. x x x p p = b

15 Pictures of Span in R 2 Draing a picture of Span{ 1, 2,..., p} is the same as draing a picture of all linear combinations of 1, 2,..., p. Span{} Span{, } Span{, }

16 Pictures of Span in R 3 Span{} Span{, } Span{u,, } Span{u,, } u u Important Een if intuition and a geometric feeling of hat Span represents is important for class. You ill use the definition of Span to sole problems on the exams.

17 Systems of Linear Equations Question Is 16 a linear combination of 2 and 2? This means: can e sole the equation x 2 + y 2 = here x and y are the unknons (the coefficients)? Rerite: x y 8 x y 8 2x + 2y = 16 or 2x 2y = 16. 6x y 3 6x y 3 This is just a system of linear equations: x y = 8 2x 2y = 16 6x y = 3.

18 Systems of Linear Equations Is 16 a linear combination of 2 and 2? x y = 8 matrix form x 2y = x y = ro reduce Conclusion: solution = x = 1 y = 9 Systems of linear equations depend on the Span of a set of ectors!

19 Span of ectors and Linear equations We hae three equialent ays to think about linear systems of equations: Summary Let 1, 2,..., p, b be ectors in R n and x 1, x 2,..., x p be scalars. 1. A ector b is in the span of 1, 2,..., p. 2. The linear system ith augmented matrix 1 2 p b, is consistent ( i s and b are the columns). 3. The ector equation x x x p p = b, has a solution. Equialent means that, for any gien list of ectors 1, 2,..., p, b, either all three statements are true, or all three statements are false.

20 Extra: So, hat is Span? To think about... 0 Ho many ectors are in Span 0? 0 A. Zero B. One C. Infinity So far, it seems that Span{ 1, 2,..., p} is the smallest linear space (line, plane, etc.) containing the origin and all of the ectors 1, 2,..., p. We ill make this precise later.

21 Extra: Points and Vectors So hat is the difference beteen a point and a ector? A ector need not start at the origin: it can be located anyhere! In other ords, an arro is determined by its length and its direction, not by its location. ( 1 These arros all represent the ector. 2) Hoeer, unless otherise specified, e ll assume a ector starts at the origin: e ll usually be sloppy and identify the ector ( 1 2) ith the point (1, 2). This makes sense in the real orld: many physical quantities, such as elocity, are represented as ectors. But it makes more sense to think of the elocity of a car as being located at the car. Another ay to think about it: a ector is a difference beteen to points, or the arro from one point to another. (2, 3) ( ) 1 ( For instance, is the arro from (1, 1) to (2, 3). 1 2 (1, 1) 2)