1 / 79

MA122 Midterm

MA122 Midterm. Rosanna Lo - Coordinator Shawn Lucas – Tutor Greg Krikorian - Tutor. About Us. SOS started in 2004 Raise money for children in Latin America Many volunteer opportunities Check out our website: http://www.studentsofferingsupport.ca/ Video!. Shawn Lucas.

kadeem
Download Presentation

MA122 Midterm

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MA122 Midterm Rosanna Lo - Coordinator Shawn Lucas – Tutor Greg Krikorian - Tutor

  2. About Us • SOS started in 2004 • Raise money for children in Latin America • Many volunteer opportunities • Check out our website: http://www.studentsofferingsupport.ca/ • Video!

  3. Shawn Lucas Program: BBA w/Math & Comp Sci Minor Year: Second Hometown: Waterdown Favourite Team: Blue Bots (Obviously..) Qualifications: Taken all first year math courses; A+ in all Love: Corny Math Jokes

  4. Greg Krikorian • 2nd year BBA/Financial Math • From Kingston/Pittsburgh/Vancouver/Ottawa • I.A. for MA129 • Favourite Hockey Team is the Penguins

  5. About this Session • Welcome to Linear AlgeBRO • Listen! • Participate! • Ask questions! • Yell at Us System • What does little mermaid wear? • Algae-bra

  6. Agenda • Chapter 1 – Systems of Linear Equations and Matrices • Linear Systems • Gaussian Elimination • Matrices & Matrix Operations • Algebraic Properties of Matrices • Elementary Matrices & Inverse • Matrix Inverses & Linear System • Chapter 2 – Determinants • Cofactor Expansion of Determinants • Evaluating Determinants • Properties of Determinants 6

  7. Chapter 3 – Euclidean Vector Spaces • Vectors in 2-, 3-, n-space • Norm, Dot Product, & Distance in • Study Tips

  8. Chapter 1:Systems of Linear Equations and Matrices

  9. Linear Systems of Equations • What is a Linear Equation? • of the form • What is a Linear System? • two or more linear equations using the same variables • solution must satisfy each equation in the system

  10. How do we Solve? • How do we solve a Linear System? • Can use substitution & elimination • Can draw a graph • Or.. • Augmented Matrices!

  11. Matrix Forms • Row Echelon Form (REF) • Leading 1s • Leading 1 in an upper row must be to the left of the leading 1 in the row below it • Once we get into this form, can use back substitution 11

  12. Matrix Forms (ctd.) • Reduced Row Echelon Form (RREF) • In row echelon form and • All entries except leading 1s are zero • Once we get into this form, can use Gauss-Jordan Elimination 12

  13. Using Matrices to Solve • Elementary Row Operations • Multiply a row by any nonzero constant • Interchange any two rows • Replace a row by itself plus a multiple (+ or -) of another • Caution: DO NOT combine the first and last; it’s a shit storm • Gaussian Elimination • Step 1: Get leading 1 in first row • Step 2: Get zeroes for all entries below the first leading 1 • Step 3: Repeat Process with second row, and so on

  14. Using Matrices to Solve (ctd.) • Gaussian Elimination • Step 1: Get leading 1 in first row • Step 2: Get zeroes for all entries below the first leading 1 • Step 3: Repeat Process with second row, and so on • Now we have REF, but we need RREF: • Use EROs to introduce zeroes above leading 1s • ERO = Elementary Row Operations

  15. Example • Solve the system of equations

  16. Solution • Step 1: Get leading 1 in first row • Step 2: Get zeroes for all entries below the first leading 1 • Step 3: Repeat Process with second row, and so on Now we have REF, but we need RREF!!

  17. Example • Solve the system of equations • Solution:

  18. Matrix Operations • Addition/Subtraction • Two matrices A and B can only be added if: • They are the same size • How does this work? • Add numbers in same position in each matrix 18

  19. Example • Find the matrix C, where C = A + B, given the following matrices: A = B = 19

  20. Solution C = A + B C = + C = 20

  21. Matrix Operations • Scalar Multiplication • The matrix A can also be multiplied by a constant, say c • Multiply each entry in the matrix by c • Quick Example: • Find the matrix cA if A = and c = 2 • Solution: 21

  22. Matrix Operations • Matrix Multiplication • Suppose we have two matrices, A and B, and we wish to multiply them to form a new matrix AB.. • Important: The number of columns in A must equal the number of rows in B • e.g is 2 x 3, and so it can only be multiplied by a matrix that is 3 x n: “The entry of the jth row and kth column of the new matrix AB is the dot product of the jth row of A with the kth column of B.” = 22

  23. Example • Find the matrix C = AB if A = B = C = 23

  24. Matrix Operations • Transpose • For a matrix A, the transpose is denoted AT • How? Turn rows into columns and columns into rows • E.g. Find the transpose of • Solution: Rotate 45° clockwise and then swap columns OR; Your R1 is now C1, R2 is now C2, etc. - Solution: 24

  25. Matrix Operations • Trace • tr(A) • sum of the entries on the main diagonal of A 25

  26. Algebraic Properties • Inverse • The matrix A has an inverse matrix B if: • BA = AB = I • Recall, I is the identity matrix: • How do we find the inverse? • Row reduce the matrix [ A | I ] until the left part is in RREF • You will end up with a matrix of the form [ I | B ] where B = A-1 • Note that if the RREF of A is not I, then A is not invertible 26

  27. Example • Determine whether A = is invertible, and if so, determine its inverse.

  28. Solution R1-R2 R2-R1 R3-2R2 R3-2R1 R1+2R3 -R3

  29. Example • Determine whether A = is invertible, and if so, determine its inverse. • Solution: • Check: =

  30. Algebraic Properties • Using the Inverse to Solve a System • Suppose we have a system of equations Ax=b • E.g. • If we multiply both sides of the equation Ax=b by A-1, we get: A-1 Ax= A-1 b Ix= A-1 b x= A-1 b • So now we have another way to solve for x besides row reducing!

  31. Example • Solve the system of equations x1+x2+2x3 = 2 x1+2x2+2x3 = 7 2x1+4x2+3x3 = 5

  32. Solution • Step 1: Put into a Matrix • Step 2: Find the Inverse A-1(we found this already in the last example) • Step 3: Multiply the Inverse by b A= b= =

  33. Example • Solve the system of equations x1+x2+2x3 = 2 x1+2x2+2x3 = 7 2x1+4x2+3x3 = 5 • Solution: x1 = - 21, x2 = 5, x3 = 9

  34. Algebraic Properties • Matrix Polynomials • For a given polynomial , we define the matrix polynomial in Ato be • In other words, we substitute A for x and replace a0 with a0I • In better words; Change x to A (input matrix A), and any constants in your equation change to the identity matrix

  35. Example • Given and A = find p(A). • Solution:

  36. Chapter 2:Determinants

  37. The Determinant • What is the determinant? • For a 2x2 matrix: • det(A) = a11a22-a12a21 • For a 3x3 matrix: • det(A) = ai1Ci1+ai2Ci2+ai3ci3 “Cofactor Expansion” • Where C is the cofactor

  38. Cofactor • What is the cofactor? • First, some notation: • Let A(j,k) be the 2x2 matrix obtained from A by deleting row j and column k • Then the cofactor Cjk = (-1)(j+k)·det[A(j,k)] • Checkerboard method on board

  39. Example • Find the determinant of .

  40. Solution • Recall two important formulae: • det(A) = ai1Ci1+ai2Ci2+ai3ci3 • Cjk = (-1)(j+k)·det[A(j,k)] • Choose to expand along the last row: • det(A)= 1·C31+6·C33 • det(A)= 1·(-1)4·det[A(3,1)] +6·(-1)6·det[A(3,3)] • det(A)= • det(A)= (-1)(5)-(3)(1) + 6[(4)(3)-(2)(-1)] • det(A)= 76

  41. Properties of Determinants • Adjoint • Create a matrix of the cofactors for each number in the original matrix • Find the transpose of this matrix Original Matrix A Matrix of Cofactors Transpose

  42. Example • Find the adj(A) for A = • Solution: • The cofactors are C11 = -6 C12 = -3 C13 = 20C21 = 8 C22 = 4 C23 = -10 C31 = -8 C32 = 21 C33 = -40

  43. Solution • Solution: • The cofactors are Transpose C11 = -6 C12 = -3 C13 = 20C21 = 8 C22 = 4 C23 = -10 C31 = -8 C32 = 21 C33 = -40

  44. Properties of Determinants • We can use the adjoint to find the inverse of a matrix:

  45. Example • Find the inverse of A = • We know adj(A) from the last example…what’s det(A)? • Recall, we found C21 = 8, C22 = 4, C23 = -10 • Using this we know (Cofactor expansion middle)det(A) = (6)(8) + (8)(4) + (3)(-10)det(A) = 50 • And so…

  46. Solution

  47. Cramer’s Rule • Suppose we have a system of linear equations Ax = b… • Also suppose det(A) is not zero… • Then, we know the system has a unique solution, and that solution is: • What is An? It’s the matrix A, except that we substitute the entries in the nth column with the entries in b

  48. Example • Use Cramer’s rule to solve the system of equations:x – 4y + z = 64x – y + 2z = -12x + 2y – 3z = -20 • Step 1: Put into a matrix A = B =

  49. Solution A1 = A2 = A3 =

  50. Solution

More Related