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Math 40 Lecture 1: Vectors and The Dot Product: Question 1. Given Two Points (X

This document provides an introduction to vectors and the dot product in linear algebra. It defines what vectors are in Rn as strings of n real numbers representing points. It introduces vector addition by adding corresponding components and scalar multiplication by multiplying each component by a scalar. The dot product of two vectors is defined as the sum of the products of their corresponding components, producing a scalar rather than a vector. The dot product can be used to define the length of a vector and relationships like the Cauchy-Schwarz inequality. Geometric interpretations of these linear algebraic concepts are discussed.

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53 views3 pages

Math 40 Lecture 1: Vectors and The Dot Product: Question 1. Given Two Points (X

This document provides an introduction to vectors and the dot product in linear algebra. It defines what vectors are in Rn as strings of n real numbers representing points. It introduces vector addition by adding corresponding components and scalar multiplication by multiplying each component by a scalar. The dot product of two vectors is defined as the sum of the products of their corresponding components, producing a scalar rather than a vector. The dot product can be used to define the length of a vector and relationships like the Cauchy-Schwarz inequality. Geometric interpretations of these linear algebraic concepts are discussed.

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narendra
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© © All Rights Reserved
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MATH 40 LECTURE 1: VECTORS AND THE DOT PRODUCT

DAGAN KARP

Our goal in this course is to begin a study of the beautiful world of the linear linear
objects, linear operators, their algebra and even their geometry. Said differently, we hope
to study some of the algebraic properties of Rn .
The real numbers have a familiar algebraic structure. Given two real numbers a and b,
we understand a + b, a b, a b and a/b (so long as b 6= 0 for the latter).
Question 1. Given two points (x1 , x2 , . . . , xn ) and (y1 , y2 , . . . , yn ) in Rn , can I add them? Can
I multiply them?
The answer is yes! Sort of. Well, kind of. We really need to work this out in some detail.
To have this discussion its useful to build some additional terminology. Lets begin with
vectors.
Definition 1. A vector ~v in Rn is simply a point in Rn .
Remark 2. We denote the components of ~v by ~v = (v1 , v2 , v3 , . . . vn ). In other words, a
vector in Rn is a string of n real numbers.
Why in the world do we call them vectors instead of points? Well, one answer is geometric. I think of points as zero dimensional geometric objects. The point (1, 2, 3) R3 is
just a dot, sitting at the location (1, 2, 3). However, I think of the vector (1, 2, 3) as an arrow
starting at the origin, ending at (1, 2, 3). This vector has magnitude and direction.
z

y
x
F IGURE 1. The green point p = (1, 2, 3) and the grey vector ~v = (1, 2, 3).
Adding two vectors seems easy enough. We simply add componentwise. If a
~ =
~
(a1 , . . . , an ) and b = (b1 , . . . , bn ), then
a
~ + ~b = (a1 + b1 , . . . , an + bn ).
Date: January 17, 2012.
These are lecture notes for HMC Math 40: Introduction to Linear Algebra and roughly follow our course
text Linear Algebra by David Poole.
1

Remark 3. What is the geometric interpretation of vector addition?


We may multiply vectors using the dot product.
Definition 4. The dot product of a
~ and ~b is defined by
a
~ ~b = a1 b1 + a2 b2 + + an bn .
Remark 5. Note that a
~ ~b is a simple real number, not a vector.
Definition 6. To distinguish between elements of R and vectors in Rn , we call elements of R
scalars.
Notice that we can also multiply scalars and vectors. This is creatively termed scalar
multiplication.
Definition 7. Let c be a scalar and ~v be a vector. The scalar multiple of ~v by c is
c~v = (cv1 , cv2 , . . . , cvn ).
Remark 8. What is the geometric interpretation of scalar multiplication? Does this shed
light on the term scalar?
Theorem 9. Let ~u, ~v and w
~ be vectors in Rn and let c be a scalar.
(1)
(2)
(3)
(4)

~v w
~ =w
~ ~v
~u (~v + w
~ ) = ~u ~v + ~u w
~
(c~u) ~v = c(~u ~v)
~u ~u > 0 and ~u ~u = 0 if and only if ~u = 0

P ROOF. Use the definitions.

So, this is the dot product. Hooray! Why should we care about it? Is it useful? Does it
yield any information?
Definition 10. The length of ~v is defined by
q

||~v|| = ~v ~v = v21 + + v2n .


Theorem 11. Let ~v be a vector in Rn and let c be a scalar. Then

(1) ||~v|| = 0 if and only if ~v = 0 .


(2) ||c~v|| = |c| ||~v||.
Theorem 12 (Cauchy-Schwarz Inequality). Let ~u and ~v be vectors in Rn . Then
|~u ~v| 6 ||~u|| ||~v||.
Theorem 13 (The Triangle Inequality). Let ~u and ~v be vectors in Rn . Then
||~u + ~v|| 6 ||~u|| + ||~v||.
P ROOF. Study ||~u + ~v||2 and use Cauchy-Schwarz.
Definition 14. The distance between ~u and ~v is ||~u ~v||.
Theorem 15. For nonzero vectors ~u and ~v, let be the angle between ~u and ~v. Then
~u ~v
cos =
.
||~u|| ||~v||
2

~u ~v
~v

~u

F IGURE 2. The geometry of ~u ~v


P ROOF. Apply the law of cosines to Figure 2, to obtain
||~u ~v||2 = ||~u||2 + ||~v||2 2||~u|| ||~v|| cos .

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