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u ~

w ~ α = α~ u + (1 − α)~v ~v

Jason Siefken

Inquiry Based Linear Algebra  c Jason Siefken, 2016–2018 Creative Commons By-Attribution Share-Alike

About the Document This document is a hybrid of many linear algebra resources, including those of the IOLA (Inquiry Oriented Linear Algebra) project, Jason Siefken’s IBLLinearAlgebra project, and Asaki, Camfield, Moon, and Snipes’ Radiograph and Tomography project. This document is a mix of student projects, problem sets, and labs. A typical class day looks like: 1. Introduc the day’s 2. Students problem. that stud 3. Instructo instructo to move has unde computa If studen guidance 4. Repeat s Using this form especially prim into each probl This problem-s focus on simple License Unless Commons By-A modify this doc and you releas http://creat If you modify this document, you may add your name to the copyright list. Also, if you think your contributions would be helpful to others, consider making a pull requestion, or opening an issue at https://github.com/siefkenj/IBLLinearAlgebra Content from other sources is reproduced here with permission and retains the Author’s copyright. Please see the footnote of each page to verify the copyright.

Task 1.1: The Magic Carpet Ride You are a young traveler, leaving home for the first time. Your parents want to help you on Hands-on experience with vectors as your journey, so just before your departure, they give you two gifts. Specifically, they give you displacements. two forms of transportation: a hover board and a magic carpet. Your parents inform you that  Internalize vectors as geometric objects representing displacements. both the hover board and the magic carpet have restrictions in how they operate:  Use column vector notation to write

vectors.   3  Use pre-existing knowledge of algebra We denote the restriction on the hover board’s movement by the vector . 1 to answer vector questions. By this we mean that if the hover board traveled “forward” for one hour, it would move along a “diagonal” path that would result in a displacement of 3 miles East and 1 mile North of its starting location.

  1 . 2 By this we mean that if the magic carpet traveled “forward” for one hour, it would move along a “diagonal” path that would result in a displacement of 1 mile East and 2 miles North of its starting location. We denote the restriction on the magic carpet’s movement by the vector

Scenario One: The Maiden Voyage Your Uncle Cramer suggests that your first adventure should be to go visit the wise man, Old Man Gauss. Uncle Cramer tells you that Old Man Gauss lives in a cabin that is 107 miles East and 64 miles North of your home. Task: Investigate whether or not you can use the hover board and the magic carpet to get to Gauss’s cabin. If so, how? If it is not possible to get to the cabin with these modes of transportation, why is that the case?

Drawings by @DavidsonJohnR (twitter)

1

 c IOLA Team iola math vt edu

Task 1.2: The Magic Carpet Ride, Hide and Seek You are a young traveler, leaving home for the first time. Your parents want to help you on your journey, so just before your departure, they give you two gifts. Specifically, they give you two forms of transportation: a hover board and a magic carpet. Your parents inform you that both the hover board and the magic carpet have restrictions in how they operate:

Address an existential question involving vectors: “Is it possible to find a linear combination that does. . . ?” The goal of this problem is to  Formalize geometric questions using the language of vectors.

   Find both geometric and algebraic 3 . arguments to support the same con1 clusion. By this we mean that if the hover board traveled “forward” for one hour,  Establish what a “negative multiple” it would move along a “diagonal” path that would result in a displacement of a vector should be. of 3 miles East and 1 mile North of its starting location. We denote the restriction on the hover board’s movement by the vector

  1 . 2 By this we mean that if the magic carpet traveled “forward” for one hour, it would move along a “diagonal” path that would result in a displacement of 1 mile East and 2 miles North of its starting location. We denote the restriction on the magic carpet’s movement by the vector

Scenario Two: Hide-and-Seek Old Man Gauss wants to move to a cabin in a different location. You are not sure whether Gauss is just trying to test your wits at finding him or if he actually wants to hide somewhere that you can’t visit him. Are there some locations that he can hide and you cannot reach him with these two modes of transportation? Describe the places that you can reach using a combination of the hover board and the magic carpet and those you cannot. Specify these geometrically and algebraically. Include a symbolic representation using vector notation. Also, include a convincing argument supporting your answer.

2

 c IOLA Team iola math vt edu

Sets and Set Notation Set A set is a (possibly infinite) collection of items and is notated with curly braces (for example, {1, 2,3} is the set containing the numbers 1, 2, and 3). We call the items in a setelements .

DEFINITION

If X is a set anda is an element of X , we may write a ∈ X , which is read “a is an element of X .”

If X is a set, a subset Y of X (written Y ⊆ X ) is a set such that every element of Y is an element of X. Two sets are called equal if they are subsets of each other (i.e., X = Y if X ⊆ Y and Y ⊆ X ). We can define a subset using set-builder notation. That is, if X is a set, we can define the subset Y = {a ∈ X : some rule involving a}, which is read “Y is the set of a in X such that some rule involving a is true.” If X is intuitive, we may omit it and simply write Y = {a : some rule involving a}. You may equivalently use “|” instead of “:”, writing Y = {a | some rule involving a}.

DEFINITION

Some common sets are N = {natural numbers} = {non-negative whole numbers}. Z = {integers} = {whole numbers, including negatives}. R = {real numbers}. Rn = {vectors in n-dimensional Euclidean space}.

1

(a) 3 ∈ {1, 2, 3}. (c) 4 ∈ {1, 2, 3}.

The goal of this problem i  Become familiar with ∈ the context of sets.

True

(b) 1.5 ∈ {1, 2, 3}.

False

 Distinguish between ∈

False

(d) “b”∈ {x : x is an English letter}. (f) {1, 2} ⊆ {1, 2, 3}.

False

True

(g) For some a ∈ {1, 2, 3}, a ≥ 3. (h) For any a ∈ {1, 2, 3}, a ≥ 3. (i) 1 ⊆ {1, 2, 3}.

 Use quantifiers with se

True

(e) “ò”∈ {x : x is an English letter}.

True False

False

(j) {1, 2, 3} = {x ∈ R : 1 ≤ x ≤ 3}.

(k) {1, 2, 3} = {x ∈ Z : 1 ≤ x ≤ 3}.

False True

Practice writing sets using set-builder notation.

Write the following in set-builder notation

p 2.1 The subset A ⊆ R of real numbers larger than 2.  p  x ∈R : x > 2 .

The goal of this problem is to  Express English descriptions using math notation.

2.2 The subset B ⊆ R2 of vectors whose first coordinate is twice the second.

§

  ª §   ª a 2t v~ ∈ R2 : v~ = with a = 2b or v~ ∈ R2 : v~ = for some t ∈ R b t §  ª a or ∈ R2 : a = 2b . b

DEFINITION

2

Practice reading sets an notation.

1.1 Which of the following statements are true?

Unions & Intersections Two common set operations are unions and intersections. Let X and Y be sets. (union) X ∪ Y = {a : a ∈ X or a ∈ Y }. (intersection) X ∩ Y = {a : a ∈ X and a ∈ Y }. 3

 c Jason Siefken 2015 2018

 Recognize there is more than one correct way to write formal math.  Preview vector form of a line.

3

Let X = {1, 2, 3} and Y = {2, 3, 4, 5} and Z = {4, 5, 6}. Compute

3.1 X ∪ Y

{1, 2, 3, 4, 5}

3.2 X ∩ Y

{2, 3}

3.3 X ∪ Y ∪ Z {1, 2, 3, 4, 5, 6} 3.4 X ∩ Y ∩ Z ; = {}

4

Visualize sets of vectors.

Draw the following subsets of R2 . ª §   0 for some t ∈ R . 4.1 V = x ~ ∈ R2 : x~ = t ª §   t for some t ∈ R . 4.2 H = x~ ∈ R2 : x ~= 0 ª §   1 for some t ∈ R . 4.3 D = x~ ∈ R2 : x ~=t 1

The goal of this problem is to  Apply set-builder notation in the context of vectors.  Distinguish between “for all” and “for some” in set builder notation.  Practice unions and intersections.  Practice thinking abou

V

D

~ 0

4.4 N =

§

x~ ∈ R2 : x~ = t

4.5 V ∪ H.

ª   1 for all t ∈ R . 1

H

N = {}.

V ∪ H looks like a “+” going through the origin. V ∩ H = {~ 0} is just the origin.

4.6 V ∩ H.

4.7 Does V ∪ H = R2 ?

No. V ∪ H does not contain

    1 1 . while R2 does contain 1 1

DEFINITION

Vector Combinations Linear Combination A linear combination of the vectors v~1 , v~2 , . . . , v~n is a vector

5

w ~ = α1 v~1 + α2 v~2 + · · · + αn v~n . The scalars α1 , α2 , . . . , αn are called the coefficients of the linear combination.

Let v~1 = 5.1

    1 1 , and w ~ = 2~ v1 + v~2 . , v~2 = −1 1

Practice linear combinations.

The goal of this problem is to  Practice using the formal term linear Write w ~ as a column vector. When w ~ is written as a linear combination of v~1 and v~2, what are combination.

the coefficients of v~1 and v~2 ?   3 ; the coefficients are (2, 1). w ~= 2

 Foreshadow span.

4

 c Jason Siefken 2015 2018

5.2 Is 5.3 Is

  3 3 0

  3 = 3~ v1 + 0~ v2 . 3 0 = 0~ v1 + 0~ v2 . Yes. ~   4 = 2~ v1 + 2~ v2 . Yes. 0

a linear combination of v~1 and v~2 ?

Yes.

a linear combination of v~1 and v~2 ? 0   4 a linear combination of v~1 and v~2 ? 5.4 Is 0

5.5 Can you find a vector in R2 that isn’t a linear combination of v~1 and v~2 ?

No.

    1 0 = 21 v~1 + 21 v~2 and = 21 v~1 − 12 v~2 . Therefore 0 1       0 1 a = a( 12 v~1 + 12 v~2 ) + b(21 v~1 − 12 v~2 ) = ( a+b +b =a v1 + ( a−b v2 . 2 )~ 2 )~ 1 0 b

Therefore any vector in R2 can be written as linear combinations of v~1 and v~2 . 5.6 Can you find a vector in R2 that isn’t a linear combination of v~1 ?

Yes. All linear combinations of v~1 have equal x and y coordinates, therefore w ~= not a linear combination of v~1 .

  2 is 1

  Practice formal writing. 3 h= Recall the Magic Carpet Ride task where the hover board could travel in the direction ~ 1   1 . and the magic carpet could move in the direction m ~= 2

6

6.1 Rephrase the sentence “Gauss can be reached using just the magic carpet and the hover board”

using formal mathematical language. ~ Gauss’s location can be written as a linear combination of m ~ andh. 6.2 Rephrase the sentence “There is nowhere Gauss can hide where he is inaccessible by magic carpet

and hover board” using formal mathematical language.

DEFINITION

~ Every vector in R2 can be written as a linear combination of m ~ and h. ~ and m” ~ using formal mathe6.3 Rephrase the sentence “R2 is the set of all linear combinations of h matical language. R2 = {~ v : v~ = t m ~ + sh~ for some t, s ∈ R}.

7

Non-negative & Convex Linear Combinations The linear combination w ~ = α1 v~1 + α2 v~2 + · · · + αn v~n is called a non-negative linear combination of v~1 , v~2 , . . . , v~n if α1 , α2 , . . . , αn ≥ 0. If α1 , α2 , . . . , αn ≥ 0 and α1 +α2 + · · · +αn = 1, then w ~ is called aconvex linear combination of v~1 , v~2 , . . . , v~n .

Let

  1 a~ = 1

  0 c~ = 1

 ~b = −1 1 

  0 d~ = 2

 −1 . e~ = −1 

7.1 Out of a ~, ~b, c~, ~d, and e~, which vectors are

b? (a) linear combinations of a~ and ~ ~ linear combination of a~ and b.

All of them, since any vector in R2 can be written as a

~ ~ (b) non-negative linear combinations of a~ and b? a~, ~b, c~, d. (c) convex linear combinations of a~ and ~b? a~, ~b, c~. 7.2 If possible, find two vectors u ~ and v~ so that

(a) a~ and c~ are non-negative linear combinations of u ~ and v~ but b~ is not. Let u ~ = a~ and v~ = c~. (b) a~ and e~ are non-negative linear combinations of u ~ and v~. Let u ~ = a~ and v~ = e~. 5  c Jason Siefken 2015 2018

Geometric meaning of and convex linear combinations. The goal of this problem is to  Read and apply the definition of nonnegative and convex linear combinations.  Gain geometric intuition for nonnegative and convex linear combinations.  Learn how to describe line segments using convex linear combinations.

(c) a~ and ~b are non-negative linear combinations of u ~ and v~ but d~ is not. Impossible. If a~ and ~b are non-negative linear combinations of u ~ and v~, then every non-negative linear combination of a~ and ~ b is also a non-negative linear combination of u ~ and v~. And, we already concluded that ~ d is a non-negative linear combination of a~ and ~b. ~ and v~. (d) a~, c~, and d~ are convex linear combinations of u Impossible. Convex linear combinations all lie on the same line segment, but~ a , c~, and d~ are not collinear. Otherwise, explain why it’s not possible.

Lines and Planes 8

Link prior knowledge t tion/concepts.

Let A be the set of points (x, y) ∈ R2 such that y = 2x + 1. 8.1 Describe A using set-builder notation.

§





ª

t v~ ∈ R2 : v~ = for some t ∈ R 2t + 1 §   ª ª § x t or ∈ R2 : t ∈ R ∈ R2 : y = 2x + 1 or 2t + 1 y

The goal of this problem i  Convert between y = m a line and the set-buil of the same line.  Think about lines in ter rather than equations.

8.2 Draw A as a subset of R2 . 8.3 Add the vectors a ~=

   1 −1 ~ , b= and d~ = ~b − a~ to your drawing. −1 3



d~ ~b

a~

A

8.4 For which t ∈ R is it true that a ~ + td~ ∈ A? Explain using your picture.

DEFINITION

a~ + t d~ ∈ A for any t ∈ R. We can see this because if we start at the vector a~ and the displace by t d~ , we will always be on the line A.

9

Vector Form of a Line A line ℓ is written in vector form if it is expressed as x~ = t ~d + p ~ ~. That is, ℓ = {~ d x : x~ = t d~ + p ~ for some t ∈ R}. The vector ~ for some vector d~ and point p is called a direction vector for ℓ. Practice with vector form.

Let ℓ ⊆ R2 be the line with equation 2x + y = 3, and let L ⊆ R3 be the line with equations The goal of this problem is to 2x + y = 3 and z = y .  Express lines in R2 and R3 in vector form.

9.1 Write ℓ in vector form. Is vector form of ℓ unique?

 Produce direction vectors by subtracting two points on a line.

 0 1 + x~ = t 3 −2 



 Recognize vector form is not unique.

6

 c Jason Siefken 2015 2018

The vector form is not unique, as any non-zero scalar multiple of



1

 can serve as a  −2 0 direction vector. Additionally, any other point on the line can be used in place of . 3     1 −4 is another vector form of ℓ. + For example, x~ = t 1 8     0 1    −2 3 . This is obtained by finding two points: one 9.2 Write L in vector form. = + t x~ 3 −2 when x = 0 and one when x = 1 and subtracting them to find a direction vector for L . ~ and “~ 9.3 Find another vector form for L where both “ d” p” are different from before.     1 −3 x~ = t  6 +  1. 6 1

Again, any non-zero scalar multiple of the direction vector will work for ~ d, as will any other point on the line work for p ~.

10

Intersect lines in vector f

Let A, B, and C be given in vector form by A z {  }|    1 0 x~ = t  2 +  0 3 1

B z  }|   { −1 −1 x~ = t  1  +  1 2 1

C z  }|  { 2 1 x~ = t  −1 + 1  . 1 1

10.1 Do the lines A and B intersect? Justify your conclusion.

          0 0 1 −1 −1 Yes. (0)  2 +  0 = 0 = (−1)  1 +  1 . 2 1 1 1 3

To find the intersection, if there is one, we must solve the vector equation:         0 1 −1 −1 t  2 +  0 = s  1  +  1  . 2 1 1 3

One solution is when t = 0 and s = −1.

10.2 Do the lines A and C intersect? Justify your conclusion.

No. The vector equation         1 2 0 1 t 2  +  0 = s −1 +  1 1 1 1 3 has no solutions. This is equivalent to saying that the following system of equations has no solutions: t = 2s + 1 2t = −s + 1

3t + 1 = s + 1 The third equation tells us that s = 3t , which when substituted into the first equation forces t = − 51 and therefore s = −53. However, these two numbers don’t satisfy the second equation. ~ 6= q~ and suppose X has vector form x~ = td~ + p ~ and Y has vector form x~ = t d~ + q~. Is it 10.3 Let p possible that X and Y intersect? 6 0, then X and Y will actually be the same line, since in this Yes. If q~ = p ~ + ad~ for a = case x~ = t d~ + q~ = td~ + (~ p + a ~d) = (t + a) d~ + p ~. 7  c Jason Siefken 2015 2018

The goal of this problem i  Practice computing th between lines in vector  Recognize “ t ” as a du as used in vector for when comparing lines i “ t ” needs to be replac dummy variables.

For example, the following two vector equations represent the same line.

DEFINITION

0 1  1 0  x~ = t  +

11

and

0 1 Vector Form of a Plane A plane P is written in vector form if it is expressed as

7 1 . 1 7 + x~ = t 1

7

~ x~ = td~1 + s d~2 + p ~ +s d~ + p for some vectors ~ ~. That is, P = {~ d1 and ~ d2 and point p x : x~ = t d 1 2 ~ for some t, s ∈ R} . ~ ~ d are called direction vectors for P . The vectors d1 and 2

Recall the intersecting lines A and B given in vector form by A {     }| 1 0 x~ = t 2  +  0 3 1 z

B   {  }| −1 −1 x~ = t  1  +  1  . 2 1

z

Let P the plane that contains the lines A and B . 11.1 Find two direction vectors in P .

Two possible answers are:   1 d~1 =  2 3

and

  −1 d~2 =  1 . 1

These are the two direction vectors we already know are in the plane—the ones from the two lines: Note that neither of these is a multiple of the other, so they really are two unique direction vectors in P . 11.2 Write P in vector form.

      1 0 −1 x~ = t d~1 + s d~2 + p ~ = t  2 + s  1  + 0 . 1 3 1

We already have  direction vectors, so we just needed a point on the plane. We used  two 0 the point p ~ =  0 that we already know is on line A. 1

11.3 Describe how vector form of a plane relates to linear combinations.

The vector form of a plane says that a vector x~ is on the plane exactly when it is equal ~. to any linear combination of d~1 and d~2 , plus p Another way of saying the same thing is that the vector~ x is on the plane exactly when x~ − ~p is equal to some linear combination of ~d1 and d~2 .

11.4 Write P in vector form using different direction vectors and a different point.

One possible answer:



     −1 ...