Linear Algebra. 1.1 Introduction to vectors 1.2 Lengths and dot products. January 28th, 2013 Math 301. Monday, January 28, 13

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1 Linear Algebra 1.1 Introduction to vectors 1.2 Lengths and dot products January 28th, 2013 Math 301

2 Notation for linear systems 12w +4x + 23y +9z =0 2u + v +5w 2x +2y +8z =1 5u + v 6w +2x +4y z =6 8u 4v 5w x 7y =7 11u +3v +9x + y +9z = 11 3u 2v 8w 15x +5y 6z = 45 It is going to get rather cumbersome to always have to use u, v, w or x, y, z,... when we talk about larger systems.

3 Vectors Examples of a vector in two dimensions : v = apple 2 5 or v = 3 1 or v =(4, 5) or in three dimensions bold face type is used for vector variable names v = v = v =(1, 2, 3) 1 column vector row vector

4 Vectors Entries of a general vector are denoted using subscripts : v =(v 1,v 2,v 3 ) or for a vector v in R n, we write the entries as v =(v 1,v 2,...,v i,...,v n ) Notice that the vector name is bold-face, but the scalar entries are just italicized. We might still use (x, y, z) to denote a vector, but in general it is more convenient to use subscripts

5 Vector addition Vector addition is easy. If we have vectors v and w defined as v = apple v1 v 2 and w = apple w1 w 2 then v + w is defined as new vector v + w = apple v1 + w 1 v 2 + w 2 Simply add the entries component-wise to create a new vector

6 Multiplication by a scalar We can also multiply v by a scalar c to get a new vector apple cv1 c v = cv 2 Examples : new vector v =(4, 2, 7) and w =( 1, 1, 3). Then v + w =(3, 1, 10), 3v = (12, 6, 21)

7 Linear combinations Given vectors v and w, and scalars a and b, we can form a linear combination to get a new vector a v + b w = apple av1 + bw 1 av 2 + bw 2 Show that the following can be expressed as linear combinations (a) Vector addition? (a) a = b =1 (b) (c) (d) Vector subtraction? The zero vector? Scalar multiplication? (b) a = 1, b = 1 (c) a = b =0 (d) a = c, b =0

8 Geometric interpretation We can plot vectors v = (2, 4) and w = ( points in the plane. 1, 3) as head at point (v1,v2) w v tail at (0,0) Or we can plot them as vectors. All parallel vectors of the same length are the same vector!

9 Vector addition Vector v + w =(1, 7), which we can picture geometrically by first translating v to the head of w and drawing the sum. v + w v head w tail All parallel vectors of the same length are the same vector!

10 Vector addition Or we can translate w to the head of v v + w head w v tail All parallel vectors of the same length are the same vector!

11 Subtraction? To subtract w from v, we first negate w and then add the negated vector to v. Check : v w =(3, 1) w +(v w) =v w v v w w

12 Do we need to be at the origin? v =( 7, 2) w =(5, 3) w v + w =( 2, 5) v v + w

13 Do we need to be at the origin? negate w v w w w v =( 7, 2) w =(5, 3) v vw =( 12, 1)

14 Scalar multiplication Multiplication by a scalar simply scales the length of the vector 2v v 1 3 v

15 All linear combinations? Given v and w, what are the possible linear combinations? bw u =av +bw w v av Can we get any point in the plane this way? Will this work for any two vectors v and w?

16 Linear combinations Given two vectors in 2d, can we determine if they fill only a line or the entire plane? Will u = apple 3 4 and v = apple 3 4 fill the plane? No - they only fill a line because the vectors are scalar multiples of each other. What about u = apple 3 4 and v = apple 3 1? Yes - linear combinations of u and v fill the plane.

17 In 3d? Do linear combinations of u, v and w fill a line, a plane or all of three space? u = , v = and w = A plane, since the last vector is a linear combination of the first two.

18 Vector products The dot product is defined as v w = v 1 w 1 + v 2 w 2 + v 3 w 3, in R 3 dot Example : v =(2, 3) and w =( 4, 1). Then dot product of two vectors is a number, not another vector v w = (2)( 4) + (3)(1) = 5 Example : v =(1, 3, 7) and w =(2, 0, 2). Then v w = (1)(2) + (3)(0) + (7)( 2) = 12

19 Dot product Sometimes called the inner product, or a scalar product or the Cartesian product. v w = w v Rules : (cv) w = c(v w) u (v + w) =u v + u w v v =0 implies v =0 Example u (3w v) =3u w u v

20 Length of a vector Lengths are defined naturally in terms of the dot product v =(v 1,v 2 ) ` v 2 `2 = v v 2 2 = v v v 1 Notation for length of a vector is kv k. Also called the norm of v length =kv k= p v v Vectors with length 1 are called unit vectors.

21 Length of a vector p Example : u =( 3, 4, 1) Length =kvk= p q p v v = ( 3) p =2 5 Generalizes to higher dimensions. kvk= v ux n t i=1 v 2 i Don t confuse length of a vector with the number of entries in a vector.

22 Law of Cosines? b c a c 2 = a 2 + b 2 2ab cos

23 Law of Cosines Generalization of the Pythagorean Theorem u u v v ku vk 2 =kuk 2 + kvk 2 2 kukkvk cos Written using the dot product, we get (u v) (u v) =u u + v v 2 kukkvk cos u u 2(u v)+v v = u u + v v 2 kukkvk cos u v =kukkvk cos

24 Perpendicular vectors u v =kukkvk cos The dot product of two perpendicular vectors is 0. Example : u =(3, 5) and v =(5, 3)! u v =0 The dot product is not like real multiplication because the product of two non-zero vectors can be zero.

25 Using the dot product Allows us to define the angle between two vectors in any space dimension! cos = u v kukkvk What is the angle between u = (3, 4, 1) and v = (2, 0, 1)? u v = cos 1 kukkvk 5 = cos 1 p p 26 5 r! 5 = cos o

26 Using the dot product The equation of a line in standard form can be written as a dot product : ax + by = c! (a, b) (x, y) =c (x, y) (a, b) Points on the line are all the points (x,y) whose dot product with (a,b) is c.