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Chapter 5 text We will cover Sections 1 – 3 as efficiently as we can and then focus on 5.4 and 5.5. We will not cover 5.6. 5.1 Vectors as Oriented Line Segments A vector is a line segment that has a head and a tail – it points in a certain direction! We indicate which is which very carefully. AB is not the same vector as BA AB at all. Let’s sketch these. PQ QP Let’s discuss this a bit. As long as you maintain the direction and length, you may move a vector anywhere in the plane..using isometries – which ones?. Vector Equality! A different notion than set equality! If you say PQ RS , you are saying that the two vectors are PARALLEL and of the SAME LENGTH. You could, for example turn the summit and base of a parallelogram into vectors and they would be equal. You are NOT claiming they are the same point set. Note that this is NOT the same thing as claiming two triangles are equal! It’s more like a two part congruence claim. 1 Now once you have several vectors and a scalar *(1x1 matrices, real numbers)* or two you can add, subtract, and multiply. Vectors add “head to tail”…move the second one carefully over to the head of the first one, snap them together and sketch in the resultant vector Let’s take V1 as given and one that points in a different direction V2, and add them “head to tail” V1 + V2. Let’s do one together! (0, 2) to (5, 4) added to (7,1) to (9, 2) 2 Assignments! Then do the Adding Tip to Tail one on your own. Remember to discuss LENGTH and Orientation A little Trig Review! Now there is a zero vector and vectors other than zero have inverses – Given V1 – let’s pick coordinates together: tip tail what is inverse V1under addition? What does it look like? How can we write it – two ways! What do the words “inverse V1” remind you of…starts with a G and is one syllable! How would I write: double the length of V1? Show these on the graph on the next page: 3 The zero vector is a point and has length zero: AA . The length of a vector is it’s “norm” and is designed by absolute value bars: AB Also called the magnitude. Given that we are working in the Cartesian Plane what do you suppose the length formula is? 4 Scalar multiplication is shown by multiplication: 2 AB is double the original length.***Is this like a similarity transformation? So vectors have direction. They also have LENGTH if they’re not the zero vector. If we have vector AB , and you want the length, write this AB (called “ the magnitude of vector AB”). Sketch the vector with A = (1, 1) as its tail and B = (5, 3) as its tip. Call it AB . Find the magnitude: AB What is the angle of elevation of this vector? Find − AB 5 Find 2 AB …this is called scalar multiplication Check the magnitude or length! Find a vector with its tail at the origin that has the SAME magnitude and call it OP Is it true that AB = OP ? See page 163 second paragraph for this very unusual definition of “equals”! Do ASSIGNMENTS! New Vector, negative Vector, Magnitude now. Do the zero vector Assignment, too. 6 Set up the vector from (1, 2) to (3, 3). Find 3 equal vectors – share the point coordinates with your neighbors and me – I’ll list some on the board. One last ASSIGNMENT on adding vectors #2. Any questions so far? 7 5.2 Representing Vectors with Coordinates Now back to vectors in the Cartesian Plane Given V1 (0, 0) to (4, 6)… AB … we will first note that this is a nice vector and it has a magnitude and an angle of elevation which are: 8 Now let’s look at another way to express it: 4 6 Note, too, that it is not unique…the vector from (1, 1) to (5, 7) equals this vector and has the SAME length and angle of elevation! It just has a different TAIL! The new “column” vector notation above makes it EASY to add, subtract, and multiply by scalars. 1 3 Given A = and B = find OA − OB , sketch the resultant vector. 5 6 Now do the work with the column vectors on the next page…do you get the same resultant vector? 9 Work Now let’s look for just a moment at using vector subtraction to find a straight forward representation of a vector. Let’s use the ones from above: 1 3 Given A = and B = 5 6 Let’s look at A and B individually and then look at AB , what are the point coordinates of the resultant AB ? 10 OA + AB = OB put the superscript arrows on So that AB = OB – OA put the superscript arrows on (head minus tail) Now do the ones in the ACTIVITIES: More practice, with algebra and sketches Representing vectors as Coordinates 11 5.3 Matrices Using row and column vectors has taken on a life of its own with entire books devoted to the intricacies and theorems of matrix work. We’ll just look at it all with an eye toward its application to geometry. A matrix is an array of numbers arranged as a rectangle (recall, squares are proper subsets of rectangles!). We mention the height or number of rows and then the width or number of columns. A 3 x 2 matrix has 3 rows and 2 columns. Each entry in the matrix has its own designation as a double subscript ( i for row and j for column). Let’s write out a 3 x 2 matrix and give the place names to each entry in the array a ij = Two matrices are said to be equal if they are the same size and each entry in the array is the same in both. Let’s take a minute and review: Equal line segments vs congruent line segments Equal vectors and Equal matrices 12 Adding, subtracting, and scalar multiplication is very direct. You can only add and subtract matrices that are exactly the same size, else you say that “the operation is not defined” 5 1 3 is not defined 1 5 3 1 1 6 4 is defined and has an answer x Now VECTORS are columns. The vector AB y 2x is scalar multiplication 2 y 2 AB Let’s create two 3 x 2 matrices and then add and subtract them A B A+B A–B 13 NOW – matrix multiplication is a VERY different thing than most of you would imagine. For one thing it is NOT commutative (where have we seen this before!) And for another it is highly structured. There are MANY instances in which multiplication is not a defined operation. Let’s look at a schematic for telling if it is defined: Look at the rows x columns for each matrix 2 x 2 times 3 x 2 is 3 x 1 times 1 x 5 is Now let’s do it 2 x 2 times a 2 x 1 = 2 x 1 1 3 2 1 5 3 Let’s review the schematic on how to tell BEFORE you start work, if the operation is defined or not: rows x columns times rows x columns 14 Now let’s do another one 3 x 2 times 2 x 3 – let’s pick the elements in the array What size will the answer be? 15 5 What does the vector 2 look like in 3D? 2 Do the Assignments: how many dimensions now. Include your BEST sketch! What kinds of matrices can we multiply this 3x1 by? Pick a 1 x 5 and multiply them. 3x1 times 1x5 = 3x5 Does it work the OTHER way? 1x5 times 3x1? 16 Together now: Write out a blank 3x3 matrix and fill in the locations with the location names using aij notation. Do A23 in Assigments now 17 Now let’s look at two 2 by 2’s and multiple them AB then BA. It doesn’t commute! Let’s pick the numbers together: Matrix A Matrix B AB = BA = Now do It Doesn’t Commute in ACTIVITIES. 18 Now 2 x 2 matrices with inverses together with matrix multiplication make a group. Let’s look at the identity matrix and check that it makes no changes when used with matrix multiplication. Check it on BOTH sides of matrix A 19 AND here’s a fairly nice application of matrices, as an aside before we get inverses. Given 2x – 2y = –2 3x + 1y = 5 Solve for x. Today we will learn how to set up solving for x and y by Gaussian elemination First make a 2 x 2 matrix of the coefficients. Then a column matrix of the variables and then a matrix of the right hand sides of the equations. Then check to see that it is actually the system of equations! Then get the identity matrix (carefully!) on the left and read off the answer. 20 Let’s do this again: x+y=2 2x – 3y = –11 21 Another one? Let me show you how I build them! Even more good ones: 3x + 2y = 1 2x – y = 6.5 x+y=2 2x – 3y = 9 22 Now together let’s look at CX = D 1 2 x 3 2 3 y 5 Find the x and y that make this true. What are the pieces…first reconstruct the original problem and then go from there! Go to ACTIVITIES and write the problem in Matrix form. 23 And coming right up we’ll need to know about the transpose of a matrix (in 5.5). If A is a 1 x 3 matrix then A transpose is a 3 x 1 matrix. You interchange columns and rows. You also write “A transpose” with a superscripted “T”. Let’s build a 2 x 2 matrix and it’s transpose A 3 x 4 matrix and it’s transpose 24 25 Chapter 5, section 4 and the Group of Invertible 2x2 Matrices Let’s take a minute to review some details of Mathematical Groups. You need a set and a combining operation. You need closure, an identity, an inverse for each set element. You need for the operation to be associative. We’re going to make a careful choice of set. We’re going to choose 2x2 matrices that are “invertible” and matrix multiplication. Invertible means that the matrix has an inverse! How do you know that a matrix has an inverse? Simple but not easy: The determinant is not zero. Backing up, what’s a determinant? It’s a single number computed according to (what else?) the determinant formula. If it is NOT zero, then the matrix at hand has an inverse. What is the determinant formula? For a 2x2 matrix a b c d the determinant is: ad – cb Let’s try one together: You pick the entries! Go to ACTIVITIES B and do the DETERMINANT now 26 Of course it’s not so simple for a 3x3 matrix a d g b e h c f i Write the steps here: Go to ASSIGNMENTS and do the 3x3 Det Now 27 So we’re picking 2x2 matrices with the determinant not zero. We’re picking to have inverses! Now if you multiply a 2x2 times a 2x2 do you get a matrix in our set? This covers Closure. Who knows the 2x2 identity matrix? Go to ACTIVITES and demonstrate that it IS the Identity! What is the determinant of the Identity Matrix? Is it in our set? 28 What about FINDING the Inverse of an invertible matrix. Well there’s an algorithm…a series of steps to follow. Create a 2x 4 augmented matrix. Use row operations to reveal the Id on the left and the inverse on the right. What row operations are allowed? Exchanging rows, Adding rows and multiples of rows. 4 2 3 1 First: is this invertible? Next let’s make an augmented matrix. Then let’s solve it for the inverse and then TEST to see if what we’ve got is the inverse! For the homework: how would you handle a 3x3 to find the inverse? 29 More work. Check on both sides! Go to ACTIVITIES B and do this on your own now! Pick at least TWO matrices in your group to work on. 30 Now for the killer project. Go to ACTIVITIES and illustrate associativity. So now let’s review and see if we have a group. And discuss how we know we have a group. 31 Now to tie matrices back to our earlier work in Euclidean Geometry. See page 179 in the book: Functional form: Show 2 right and 3 down as functional instructions F( ) Show it in equation form x’ = x + 2 y’ = y – 3 Now in matrix form: X’ = X + H where X’ and X are 2x1 matrices and H is a 2x 1: Let’s move a triangle using the matrix form. (1, 1), (3, 4), and (5, 1) is our triangle. Where do we want to move it? Graph paper is on the next page: 32 Is this an isometry? What kind of isometry? Let’s look at a Similarity Transform – restricted to stretching and shrinking along a major axis F(x, y) = F( ax, by) How can we show this with matrices (not in the BOOK!) 33 Transformations taking lines to lines. Page 180, text Matrix Form X’ = AX + H Where A is a 2x2 invertible matrix. Let’s create one of these and use it on the line segment (1, 1) to (3, 4). Graph paper is on the next page 34 35 A note on determinants. If you transform a triangle or closed shape using a matrix, the original area becomes the new area by multiplying the original area by the determinant of the matrix. Let’s illustrate this: 36 Go to Activities and finish up! 37