Ex 3: 5.01,5.08,6.04,6.05,6.06,6.07,6.12

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1 Advanced Math: Linear Algebra Overview Ex 3: 5.01,5.08,6.04,6.05,6.06,6.07,6.12 Exeter 3 We will do selected problems, relatively few and spread out, primarily as matrices relate to transformations. Haese & Harris High Level Second Edition 10/22 Overview 1

2 Exeter 3 Ex 3: 5.01,5.08,6.04,6.05,6.06,6.07, /22 Ex 3 p

3 Exeter 3 Ex 3: 5.01,5.08,6.04,6.05,6.06,6.07, /22 Ex 3 p

4 Objectives 1. Understand matrix structure and meaning. 13A: #1,4 (Matrix Organization) 13B.1: #1d,2d,5,8 (Add subtract) 13B.2: #2d,4,5 (Scalar multiply) 13B.3: #1hi,2hi,3c (Operations) We can organize the information into tables (in several ways): This Month Last Month Small Large Small Large Walnut Pine Cherry Walnut Pine Cherry Small Racks Large Racks Last Month This Month Last Month This Month Walnut Pine Cherry Walnut Pine Cherry A table can be written as a "matrix" consisting of just the relevant numbers. Matrix Vocabulary Rows Columns Column Dimension (= 2) Elements General Element a ij is in the i th row, j th column Row Dimension (= 3) This Month (A) Last Month (B) General Matrix Dimensions: Row x Column = 3 x 2 Matrices (plural) aka Order of a matrix Note that the rows and columns can be considered as vectors, more specifically as row vectors and column vectors. 10/24 13A: Intro to Matrices 4

.1.2.3 Objectives 1. Understand and use addition, subtraction and scalar multiples of matrices. It's not hard, but you have to pay attention!

5 Objectives 1. Understand and use addition, subtraction and scalar multiples of matrices. It's not hard, but you have to pay attention! Questions: Try to define a matrix algebra to enable you to write the answers to these questions using algebraic notation. Let A be the sales for this month, and B be the sales for last month. Define your operations formally using a general matrix. Identify any conditions that are required for each operation to exist. 1. What would you do to find the sales of both months combined? 2. What would you do to find the change in sales from last month to this month? 3. What would you do to find the average monthly sales for the two months? 4. What would you do to find the total annual sales if this month's sales were repeated every month for 12 months? 5. What would you do to predict sales 6 months from now if the change in sales from last month to this month continues? An example of what I'm looking for in response to each of the above: Answer the question in words Find the requested values 1. To find the combined sales of both months, add the matrices together. Small Large Walnut Pine Cherry Define the general case General Matrix Addition Conjecture about and summarize the properties of the operation(s) involved. Definition Constraints Commutative? Properties of Matrix Addition A + B = C means c ij = a ij + b ij Dimensions of A and B must be the same. Dimensions of C will be the same as A and B Yes Associative? Identity? Inverse? Yes A + O = A A + ( A) = O 13A: #1,4 (Matrix Organization) 13B.1: #1d,2d,5,8 (Add subtract) 13B.2: #2d,4,5 (Scalar multiply) 13B.3: #1hi,2hi,3c (Operations) 10/24 13B.1 B.3: Basic operations 5

6 What would you do to find the change in sales from last month to this month? Notice the order of subtraction! Last Month (B) This Month (A) Change (A B= C) General Subtraction A B = C means c ij = a ij b ij Question: Can you subtract any two matrices? Answer: NO! To subtract matrices, their dimensions must be the same! 13B1: Subtraction 6

7 What would you do to find the average monthly sales? First find the total: This Month (A) Last Month (B) Total (A + B = C) Then divide by 2 (times.5) A + B Average = (A + B)/2 General multiplication by a scalar ka = B means b ij = ka ij Question: Can you multiply any matrix by any scalar? Answer: YES! 13B2: Scalar Multiple 7

8 What would you do to find the total annual sales if this month's sales were repeated every month for 12 months? What would you do to predict sales 6 months from now if the change in sales from last month to this month continues? 12A = Annual sales if this month's sales continue. A + 6(A B) = Prediction for 6 months from now if changes continue You can use matrices to represent multiple equations: A B C Can be written as: 4A + B = C 13B.3: Scalar multiple and notation 8

9 Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.7,58.8,59.3,59.5 Exeter 2 10/26a Ex 2 p

10 Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.7,58.8,59.3,59.5 Exeter 2 10/26b Ex 2 p

11 Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.7,58.8,59.3,59.5 Exeter 2 10/26c Ex 2 p

12 Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.7,58.8,59.3,59.5 Exeter 2 10/26d Ex 2 p

13 Exeter 3 Ex 3: #13.02,13.03,13.04,14.02,14.03,14.08,15.05,15.09 (Matrix intro) 10/26e Ex 3 p

14 Exeter 3 Ex 3: #13.02,13.03,13.04,14.02,14.03,14.08,15.05,15.09 (Matrix intro) 10/26f Ex 3 p

15 Exeter 3 Ex 3: #13.02,13.03,13.04,14.02,14.03,14.08,15.05,15.09 (Matrix intro) 10/26g Ex 3 p

13A: #1,4 (Matrix Organization) Take questions on all 13B.1: #1d,2d,5,8 (Add subtract) 13B.2: #2d,4,5 (Scalar multiply) 13B.3: #1hi,2hi,3c (Operations) Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.

We can organize the price information in a table as follows: Walnut Pine Cherry or as a matrix: Price 150 100 200 What calculation would you do to find the total income from small racks this month?

16 13A: #1,4 (Matrix Organization) Take questions on all 13B.1: #1d,2d,5,8 (Add subtract) 13B.2: #2d,4,5 (Scalar multiply) 13B.3: #1hi,2hi,3c (Operations) Ex 2: #53.6,54.6,55.1,55.3,55.4,55.7,56.6,58.7,58.8,59.3,59.5 Ex 3: #13.02,13.03,13.04,14.02,14.03,14.08,15.05,15.09 (Matrix intro).4 Objectives 1. Understand and use matrix multiplication 13B.4: #3,4 (Dot product) 13B.5: #3, 4, 6 (Matrix multiply) 13B.6: #2,4 (GDC multiply) Multiplying matrices Now suppose that small walnut cabinets sell for $150, small Pine cabinets sell for $100, and small Cherry cabinets sell for $200 each. We can organize the price information in a table as follows: Walnut Pine Cherry or as a matrix: Price What calculation would you do to find the total income from small racks this month? is called a row vector. (A matrix with a single row is called a vector) Let's organize the small rack sales for this month like this: It's called a...you guessed it column vector. Total income for small sales this month is the dot product of the row and column vectors..5 We can organize the total sales calculations for both months as follows: Prices Walnut Pine Cherry Months This Last Sales Walnut Pine Income This Last Cherry Row 1 times Col 1 = (150 * 95) + (100 * 316) + (200 * 205) = = Row 1 Col 1 or Element 1,1 Row 1 times Col 2 = (150 * 125) + (100 * 278) + (200 * 225) = 91550= Row 1 Col 2 or Element 1,2 To multiply two matrices, find the dot products of all the combinations of row and column vectors. General Matrix Multiplication Calculate the dot product of the i th row of the first matrix with the j th column of the second matrix to get the ij th element of the resulting product Note that to multiply two matrices: > the column dimension of the first must equal the row dimension of the second. > The result will be the row dimension of the first x the column dimension of the second. n p p m n m m x n n x p m x p must be the same! A B C Try a few.6 Objectives 1. Understand and use technology for matrix multiplication Your TI 84 is powerful for this! Define matrix Then use [2nd][MATRIX][NAMES] in calculations. 13B.4: #3,4 (Dot product) 13B.5: #3, 4, 6 (Matrix multiply) 13B.6: #2,4 (GDC multiply) 10/29: 13B.4 B.6 Multiply 16

17 UNM PNM Math contest This Sunday, 11/4 from 9 12 in the Atrium Please register at UNM PNM web site 13B.4: #3,4 (Dot product) 13B.5: #3, 4, 6 (Matrix multiply) 13B.6: #2,4 (GDC multiply) Present/discuss as needed.7 Objectives 1. Understand and use matrix properties 13B.7: #1 7 (Matrix Properties) 13B.8: #1 7 (Matrix algebra) With an understanding of matrix operations, you will now develop some properties of matrices, some of which you are already familiar with. Pay attention to the results of your work as you will want to use these properties fluidly as we move forward. 10/31 13B.7: Properties 17

The beauty of matrices is that one can manipulate large numbers

In this lesson we will practice simplifying matrix expressions

18 .8 Objectives 1. Simplify matrix expressions and equations using matrix algebra The beauty of matrices is that one can manipulate large numbers of quantities with very simple notation. In this lesson we will practice simplifying matrix expressions and equations using algebraic notation. 13B.7: #1 7 (Matrix Properties) 13B.8: #1 7 (Matrix algebra) 10/31 13B.8: Matrix algebra 18

UNM PNM Math contest This Sunday, 11/4 from 9 12 in the Atrium Please register at UNM PNM web site 13B.7: #1 7 (Matrix Properties) 13B.8: #1 7 (Matrix algebra) Discuss properties below Present #4, 7.

19 UNM PNM Math contest This Sunday, 11/4 from 9 12 in the Atrium Please register at UNM PNM web site 13B.7: #1 7 (Matrix Properties) 13B.8: #1 7 (Matrix algebra) Discuss properties below Present #4, 7. Any other questions?.1 Objectives 1. Understand the origins of the inverse of a matrix 2. Use Matrix inverses to solve a 2x2 system of equations 13C.1: #1,3,5 7,8fgh,9 (Inverses) 13C.2: #2ef,3 5 (Systems of equations) Remember that a system of linear equations can be represented as a single matrix equation. A x b means and can be written: Ax = b Like a regular equation, if we can "divide" by A we can find x. But how do we "divide" by a matrix? Think back to a scalar equation. 3x = 12. The next step can be divide both sides by 3... or... multiply both sides by Notice that the result is that we get 1 times x because is the inverse of 3. What do we multiply A by to get the matrix equivalent of 1? Well first, what is the matrix equivalent of 1? What happens when you multiply How about An identity matrix is a square matrix that when multiplied by a matrix A gives A. That is: IA = A and also AI = A An identity matrix consists of all zeroes with ones along the diagonal. 3x3 identity matrix = I 3 n x n identity matrix = I n n rows n cols The inverse of a matrix A is the matrix that multiplies A to give the identity matrix. So A 1 A = I and also AA 1 = I. But how do we calculate the inverse of a matrix? Let's start with a 2x2: Suppose we have If this is true, then AA 1 = I or This means that As simultaneous equations, we have Which we (ie. you) can solve for w, x, y, and z Inverse of a 2x2 Matrix If then Note the conditions for this to exist! If ad bc 0, we can do it. The matrix is called invertible or non singular. If ad bc = 0, we have trouble (divide by zero). The matrix is called singular and the inverse does not exist! ad bc is called the determinant of the matrix because it determines whether it has an inverse Can you show that AB = A B Determinant? of a 2x2 Matrix If then deta = A = ad bc 11/2 13C.1 Inverse of a 2x2 19

20 .2 Objectives 1. Use Matrix inverses to solve a 2x2 system of equations So back to our system: can be written as Therefore, A x b Ax = b But So... Let's look at some other examples: 13C.1: #1,3,5 7,8fgh,9 (Inverses) 13C.2: #2ef,3 5 (Systems of equations) 11/2 13C.2 Inverse of a 2x2 20

13C.1: #1,3,5 7,8fgh,9 (Inverses) Note results of 5,6 13C.2: #2ef,3 5 (Systems of equations) Present 2ef,3,5.3 Objectives 1. Understand and use Cramer's Rule 2.

3: #2 12 even (More algebra) Back in 1750, a guy named Gabriel Cramer published a 650 page summary of algebraic geometry and, in a two page appendix, included a concise explanation of the theory of

For the system we've been working with: the solutions can be found simply by calculating a ratio of determinants of suitably created matrices. Do you see what "suitably" means?

21 13C.1: #1,3,5 7,8fgh,9 (Inverses) Note results of 5,6 13C.2: #2ef,3 5 (Systems of equations) Present 2ef,3,5.3 Objectives 1. Understand and use Cramer's Rule 2. Improve matrix algebra skills, uncovering various properties. 13C.2 #2abcd (With Cramer's Rule) 13C.3: #2 12 even (More algebra) Back in 1750, a guy named Gabriel Cramer published a 650 page summary of algebraic geometry and, in a two page appendix, included a concise explanation of the theory of linear systems. In it, he described a way to easily find the solutions to linear systems without explicitly finding inverse matrices. For the system we've been working with: the solutions can be found simply by calculating a ratio of determinants of suitably created matrices. Do you see what "suitably" means? Cramer's Rule: For any general 2x2 system Try a couple yourself. Compare answers with a partner. Solve the system of equations: 2x 4y = 16 3x + 5y = 24 17x 13y = 95 13x + 11y = 19 Since you now have Cramer's rule, I've added a few problems to your HW from the last section. Easy peasy. For this section, there's not much conceptually new...just practicing algebra skills and reveal various ideas. Remember that AB BA in general...pay attention to the order of your multiplies. Try to work without going to the element expansions where possible. 13C.2 #2abcd (With Cramer's Rule) 13C.3: #2 12 even (More algebra) 11/7 13C.3 Algebra Practice 21

13C.2 #2abcd (With Cramer's Rule) 13C.3: #2 12 even (More algebra).1 Objectives 1. Understand, find, and interpret the determinant of a 3x3 matrix Calculating the determinant of a 3x3 matrix: 13D.

2: #1,2 (3x3 inverse) In General: = a[ei fh] = 2[(7)(9) (4)(6)] = 2[39] = 78 b[di fg] 3[(1)(9) (4)(8)] = 3[ 23] = + 69 + c[dh eg] + 5[(1)(6) (7)(8)] = 5[ 50] = 250 78 + 69 250 = 103...or.

22 13C.2 #2abcd (With Cramer's Rule) 13C.3: #2 12 even (More algebra).1 Objectives 1. Understand, find, and interpret the determinant of a 3x3 matrix Calculating the determinant of a 3x3 matrix: 13D.1: #1f,2,3c,5,6a,7,8 (3x3 Determinant) 13D.2: #1,2 (3x3 inverse) In General: = a[ei fh] = 2[(7)(9) (4)(6)] = 2[39] = 78 b[di fg] 3[(1)(9) (4)(8)] = 3[ 23] = c[dh eg] + 5[(1)(6) (7)(8)] = 5[ 50] = = or... (aei + bfg + cdh) (gec + hfa + idb) = (2)(7)(9) + (3)(4)(8) + (5)(1)(6) [(8)(7)(5) + (6)(4)(2) + (9)(1)(3)] = 103 Note: Yeah, calculator's are handy, however beware....2 Objectives 1. Understand, find, and interpret the inverse of a 3x3 matrix 2. Analyze the quality and rigor of a math course/text based on how early in a topic a calculator is recommended. You will explore this topic individually, by studying the topic from the book. Read Haese & Harris' entire lesson (Section 13D.2) on finding the inverse of a 3x3. Understand every detail and be certain that you understand and can replicate each step in their example. H&H Section 13D.2 (in its entirety): 13D.1: #1f,2,3c,5,6a,7,8 (3x3 Determinant) 13D.2: #1,2 (3x3 inverse) 11/9 13D.1 3x3 Determinants 22

23 Exeter 3 Ex 3: #16.06,16.07,16.08,18.01,18.05,18.08,18.09 (Transformations) 11/14 Ex 3 p

24 Exeter 3 Ex 3: #16.06,16.07,16.08,18.01,18.05,18.08,18.09 (Transformations) Ex 3: #16.06,16.07,16.08,18.01,18.05,18.08,18.09 (Transformations) 11/14 Ex 3 p

25 Exeter 3 Ex 3: #19.03,19.04,20.07,21.04,21.06,22.01,22.03,22.06,22.07 (Rotations) 11/16 Ex 3 p

26 Exeter 3 Ex 3: #19.03,19.04,20.07,21.04,21.06,22.01,22.03,22.06,22.07 (Rotations) 11/16 Ex 3 p

27 Exeter 3 Ex 3: #19.03,19.04,20.07,21.04,21.06,22.01,22.03,22.06,22.07 (Rotations) Ex 3: #19.03,19.04,20.07,21.04,21.06,22.01,22.03,22.06,22.07 (Rotations) 11/16 Ex 3 p

Complex Numbers Test #5 13D.1: #1f,2,3c,5,6a,7,8 (3x3 Determinant) 13D.2: #1,2 (3x3 inverse) Be prepared for a test after Thanksgiving, Wed 11/28 Objectives 13E: #2 10 even (Systems of equations) 1.

28 Complex Numbers Test #5 13D.1: #1f,2,3c,5,6a,7,8 (3x3 Determinant) 13D.2: #1,2 (3x3 inverse) Be prepared for a test after Thanksgiving, Wed 11/28 Objectives 13E: #2 10 even (Systems of equations) 1. Use matrix methods and Cramer's Rule to solve 3x3 systems Cramer's Rule can also be extended to larger systems. For a 3x3 system: the solutions are: (can you guess?) What happens if the determinant of the "coefficient matrix" is zero? If the determinant of the coefficient matrix = 0, the system has NO SOLUTION! Wow! Easy to find out whether there is a solution before you do all the work!...and computers can do the calculations really easily for you! Of course, with a calculator one can solve systems either using PLYSMLT2 or by finding an inverse matrix. Sometimes the trick is to coerce some real world problem into a form that allows you to use your newfound math wizardry: 13E: #2 10 even (Systems of equations) 11/20 13E: Systems 28

29 Complex Numbers Test #6 13E: #2 10 even (Systems of equations) Present and discuss as needed Objectives 1. Review and prepare for Quiz QB #1 7, QB #1 7, /26 Review 29

30 Linear Algebra Quiz 1 Matrix algebra Exeter 3 Ex 3: #23.10,23.11,24.05,25.06,25.07,26.06,26.12 (Systems) 11/28 Quiz & Ex 3 p

31 Linear Algebra Quiz 1 Matrix algebra Exeter 3 Ex 3: #23.10,23.11,24.05,25.06,25.07,26.06,26.12 (Systems) 11/28 Quiz & Ex 3 p

Ex 3: #23.10,23.11,24.05,25.06,25.07,26.06,26.12 (Systems) Questions?? Return and discuss tests.1 Objectives 1. Understand and use row operations to solve systems 13F.

7 Recall that a 2 dimensional system represents two lines that intersect, are parallel, or are the same line. More on this soon.

First, write the system in matrix form Next, create an augmented matrix by combining the coefficient matrix with the result vector: We now perform any of three valid row operations: 1) Swap rows 2)

32 Ex 3: #23.10,23.11,24.05,25.06,25.07,26.06,26.12 (Systems) Questions?? Return and discuss tests.1 Objectives 1. Understand and use row operations to solve systems 13F.1: #1 3,7,8 (2x2 Row reduction) QB #9 11 Solve the following systems of equations using elimination: 2x + 4y = 10 x 2 ( x + 6y = 1 ) 2x + 4y = 10 2x 12y = 2 + 8y = 8 y = 1 x + 6 ( 1) = 1 x 6 = 1 x = 7 Recall that a 2 dimensional system represents two lines that intersect, are parallel, or are the same line. More on this soon. We can systematize this process by putting the problem in matrix form and defining certain row operations corresponding to the algebra we just did. First, write the system in matrix form Next, create an augmented matrix by combining the coefficient matrix with the result vector: We now perform any of three valid row operations: 1) Swap rows 2) Replace a row by a multiple of the row 3) Add a multiple of one row to another Notice that the steps we took above were exactly these steps. Consider what would happen if we could do only these three things and arrive at: What does this mean? OK, here goes. Note how we keep track or our steps. This makes life easier This is row echelon form This is reduced row echelon form Alternately, we now have y = 1 which we can substitute into one of the original equations. This is done! You can show more intermediate steps if you like. We'll begin with 2x2 systems. The benefit becomes clear in 3x3 and higher. We call this "using elementary row operations"... Try one: Doing this also allows us to understand how various parameters affect systems. Recall that in 2 dimensions, we'll have one of three situations: What does row reduction look like in these cases? We've seen the first case. What about parallel lines? Interpretation? 0 = 6? No solution! Parallel! ½R1 R2 2R1 Interpretation? 0 = 0? Always! Same line! ½R1 R2 4R1 Try this: Note that we go beyond just saying "they're the same line". We introduce a parameter t to describe the relationship between x and y parametrically. 13F.1: #1 3,7,8 (2x2 Row reduction) QB # /30 13F.1 Row Reduction 32

13F.1: #1 3,7,8 (2x2 Row reduction) 2 verbal, 3,7,8 QB #9 11 Questions

Understand and use row operations on 3D systems 13F.

understand systems that do not have unique solutions (for that, our

More importantly can we be specific about the intersections when there

When a 3D system intersects in a line, we need to find the line (in

Use row reduction, then confirm with your calculator For now, we can

2: #1cd,3,5,7 (3x3 Row reduction) QB #8 An attempt to find the three

33 13F.1: #1 3,7,8 (2x2 Row reduction) 2 verbal, 3,7,8 QB #9 11 Questions as needed.2 Objectives 1. Understand and use row operations on 3D systems 13F.2: #1cd,3,5,7 (3x3 Row reduction) QB #8 Row reduction is most useful to understand systems that do not have unique solutions (for that, our calculator will work!) In 3x3 systems, we are working with planes. What are the possible solutions? Can we find which situation with matrices? More importantly can we be specific about the intersections when there is not a unique solution? When a 3D system intersects in a line, we need to find the line (in parametric form, of course). Use row reduction, then confirm with your calculator For now, we can simply say no solution. Try a more general case 13F.2: #1cd,3,5,7 (3x3 Row reduction) QB #8 An attempt to find the three equations of intersection for the second example above. The z value is problematic. Let it be a parameter t and see what happens. This is a parametric equation of a line of intersection. Is there another one? How can we find out? Swap row 1 & 2 ref(a) This is another one. But how can we find the third one? I couldn't! 12/3 13F.2 3x3 Row Reduction 33

34 13F.2: #1cd,3,5,7 (3x3 Row reduction) QB #8.3 Objectives 1. Understand and use row operations on 3D systems 13F.3: #1 4 (Over & under specified) QB #15ab 18 Let's turn our attention to the intersection of two planes. Write down the equations of two planes. Be sure they are not parallel. Represent your system with augmented matrices. Use your calculator to do row reduction. Interpret the result. The system is said to be underspecified But, we can find the equation of the line that represents the intersection. Try this one: 13F.3: #1 4 (Over & under specified) QB #15ab 18 12/5 13F.3 Under/over specified 34

35 13F.3: #1 4 (Over & under specified) QB #15ab 18.Ex Objectives 1. Find inverse of 3x3 matrix using row operations. Handout Row operations can also be used to find the inverse of a 3x3. Its not hard, but it can be tedious... Consider the matrix Begin, buy creating an augmented matrix with the 3x3 identity matrix to the right of the original matrix Now, use row operations on the entire thing to transform the original 3x3 on the left to the identity matrix. Swap R1 & R2 R2 = R2 2R1 R3 = R3 R2 R3 = R3 5 R1 = R1 2R3 The inverse matrix is Try one of your choosing. Use small numbers, but don't make it trivial. Handout 12/7 13F.Ex 3x3 Inverse byrow reduction 35

Welcome to IB Math - Standard Level Year 2.

Welcome to IB Math - Standard Level Year 2 Why math? Some things to know: www.aleimath.blogspot.com 1. Lots of info at 2. HW yup. You know you love it! Be prepared to present. Notebook all work is in it.