Announcements Wednesday, September 05

Transcription

1 Announcements Wednesday, September 05 WeBWorK 2.2, 2.3 due today at 11:59pm. The quiz on Friday coers through 2.3 (last week s material). My office is Skiles 244 and Rabinoffice hours are: Mondays, 12 1pm; Wednesdays, 1 3pm. Your TAs hae office hours too. You can go to any of them. Details on the website.

2 Chapter 3 Systems of Linear Equations: Geometry

3 Motiation We want to think about the algebra in linear algebra (systems of equations and their solution sets) in terms of geometry (points, lines, planes, etc). x 3y = 3 2x + y = 8 This will gie us better insight into the properties of systems of equations and their solution sets. Remember: I expect you to be able to draw pictures!

4 Section 3.1 Vectors

5 Points and Vectors We hae been drawing elements of R n as points in the line, plane, space, etc. We can also draw them as arrows. Definition A point is an element of R n, drawn as a point (a dot). the point (1, 3) A ector is an element of R n, drawn as an arrow. When we think of an element of R n as a ector, we ll usually write it ectically, like a matrix with one column: ( ) 1 =. 3 [interactie] the ector ( 1) 3 The difference is purely psychological: points and ectors are just lists of numbers.

6 Points and Vectors So why make the distinction? A ector need not start at the origin: it can be located anywhere! In other words, an arrow is determined by its length and its direction, not by its location. These arrows all represent the ector ( 1 2). Howeer, unless otherwise specified, we ll assume a ector starts at the origin.

7 Vector Algebra Definition We can add two 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 c: x c x c y = c y. z c z We call c a scalar to distinguish it from a ector. If is a ector and c is a scalar, c is called a scalar multiple of. (And likewise for ectors of length n.) For instance,

8 Vector Addition and Subtraction: Geometry 5 = = w w + w The parallelogram law for ector addition Geometrically, the sum of two ectors, w is obtained as follows: place the tail of w at the head of. Then + w is the ector whose tail is the tail of and whose head is the head of w. Doing this both ways creates a parallelogram. For example, ( 1 3) + ( 4 2) = ( 5 5). 5 = = Why? The width of + w is the sum of the widths, and likewise with the heights. [interactie] w w Vector subtraction Geometrically, the difference of two ectors, w is obtained as follows: place the tail of and w at the same point. Then w is the ector from the head of w to the head of. For example, ( 1 4) ( 4 2) = ( 3 2 ). Why? If you add w to w, you get. [interactie] This works in higher dimensions too!

9 Scalar Multiplication: Geometry Scalar multiples of a ector These hae the same direction but a different length. Some multiples of. ( ) 1 = 2 ( ) 2 2 = 4 1 ( ) 1 2 = 2 1 ( ) 0 0 = 0 All multiples of. [interactie] So the scalar multiples of form a line.

10 Linear Combinations We can add and scalar multiply in the same equation: w = c c c p p where c 1, c 2,..., c p are scalars, 1, 2,..., p are ectors in R n, and w is a ector in R n. Definition We call w a linear combination of the ectors 1, 2,..., p. The scalars c 1, c 2,..., c p are called the weights or coefficients. Example ( ( 1 1 Let = and w =. 2) 0) w What are some linear combinations of and w? + w w 2 + 0w 2w [interactie: 2 ectors] [interactie: 3 ectors]

11 Poll

12 More Examples ( 2 What are some linear combinations of =? 1) Question What are all linear combinations of ( ( ) 2 1 = and w =? 2) 1 w Answer: The line which contains both ectors. What s different about this example and the one on the poll? [interactie]

13 Section 3.2 Vector Equations and Spans

14 Systems of Linear Equations Sole the following system of linear equations: x y = 8 2x 2y = 16 6x y = 3. We can write all three equations at once as ectors: We can write this as a linear combination: So we are asking: Question: Is 16 a linear combination of 2 and 2? 3 6 1

15 Systems of Linear Equations Continued Conclusion: x y = 8 2x 2y = 16 6x y = 3 matrix form row reduce solution = [interactie] (this is the picture of a consistent linear system) x = 1 y = 9 What is the relationship between the ectors in the linear combination and the matrix form of the linear equation? They hae the same columns! Shortcut: You can go directly between augmented matrices and ector equations.

16 Vector Equations and Linear Equations Summary The ector equation x x x p p = b, where 1, 2,..., p, b are ectors in R n and x 1, x 2,..., x p are scalars, has the same solution set as the linear system with augmented matrix 1 2 p b, where the i s and b are the columns of the matrix. So we now hae (at least) two equialent ways of thinking about linear systems of equations: 1. Augmented matrices. 2. Linear combinations of ectors (ector equations). The last one is more geometric in nature.

17 Span It is important to know what are all linear combinations of a set of ectors 1, 2,..., p in R n : it s exactly the collection of all b in R n such that the ector equation (in the unknowns x 1, x 2,..., x p) has a solution (i.e., is consistent). x x x p p = b the set of such that 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 x 1, x 2,..., x p in R }. Synonyms: Span{ 1, 2,..., p} is the subset spanned by or generated by 1, 2,..., p. This is the first of seeral definitions in this class that you simply must learn. I will gie you other ways to think about Span, and ways to draw pictures, but this is the definition. Haing a ague idea what Span means will not help you sole any exam problems!

18 Span Continued Now we hae seeral equialent ways of making the same statement: 1. A ector b is in the span of 1, 2,..., p. 2. The ector equation has a solution. x x x p p = b 3. The linear system with augmented matrix 1 2 p b is consistent. [interactie example] (this is the picture of an inconsistent linear system) Note: 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.

19 Pictures of Span Drawing a picture of Span{ 1, 2,..., p} is the same as drawing a picture of all linear combinations of 1, 2,..., p. Span{} Span{, w} w Span{, w} w [interactie: span of two ectors in R 2 ]

20 Pictures of Span In R 3 Span{} Span{, w} w Span{u,, w} Span{u,, w} u w w u [interactie: span of two ectors in R 3 ] [interactie: span of three ectors in R 3 ]

21 Poll

22 Summary The whole lecture was about drawing pictures of systems of linear equations. Points and ectors are two ways of drawing elements of R n. Vectors are drawn as arrows. Vector addition, subtraction, and scalar multiplication hae geometric interpretations. A linear combination is a sum of scalar multiples of ectors. This is also a geometric construction, which leads to lots of pretty pictures. The span of a set of ectors is the set of all linear combinations of those ectors. It is also fun to draw. A system of linear equations is equialent to a ector equation, where the unknowns are the coefficients of a linear combination.