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# Week 6 Notes Math 340

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This 8 page Class Notes was uploaded by Susan Ossareh on Sunday February 28, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 24 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 02/28/16

Math 340 Lecture – Introduction to Ordinary Differential Equations – February 22 , 2016 nd What We Covered: 1. Quiz a. Highlights i. Euler’s Method and Matrix Algebra 2. Course Content – Chapter 7: Matrix Algebra a. Section 7.1: Vectors and Matrices i. The goal of matrix is to solve linear systems ii. Matrix: a rectangular array of numbers iii. Entries/Components: the numbers that appear in the array iv. Example: 1. For a system like 3???? + 2???? − 5???? = 5 4???? − ???? + 5???? = 0 2. We want to isolate the coefficients and put them in a matrix 3 2 −5 ???? = 4 −1 5 v. We can say a general matrix has m rows and n columns ???? 11 ⋯ ????1???? ???? = ( ⋮ ⋱ ⋮ ) ????????1 ⋯ ???????????? 1. A is a mxn matrix vi. We also have column vectors, which is a matrix with only column and row vectors which only have one row 1. Column Vector: 1 ???? = 2 3 2. Row Vector: ???? = −1 3 4 vii. Properties 1. Sum of A and B: If A and B are mxn matrices then A+B=sum “term by term” −1 0 1 2 1 + 1 0 + 2 0 2 3 5+ 2 −1 = 3 + 2 5 − 1= 5 4 2. Multiplication a. Multiplication of a matrix by a scalar is the matrix increased by the lambda ratio. If ???? ???? ℝ,???? i. ???? ∗ ???? = matrix where each component is the corresponding component from A multiplied by ???? 1 5 10 1 2( 2 ) = ( −2 −2) −1 −1 viii. Linear Combination 1. A linear combination of x and y is any vector of the form ax+by where a and b are real numbers 2. We say x is a linear combination of ???? ,…,???? . This is also a span of x 1 ???? ???? 3. Span: the span of {1 ,…,???? } vectors in ℝ is the set of all the linear combinations of those vectors a. ????1and ???? 2re linearly dependent b. Example: What is the span of {(0,1) , (1,0)}? If x is in the span, then… ???? (0 1) + ????(1 0) = ???? (0 ????) + ( ) = ???? ???? 0 (???? ????) = ???? ???????????? ???????????????? ????,???? ???? ℝ ix. Multiplication of Matrices 1. If A is an mxn matrix, B is a nxk matrix 2. If ???? = (???????????? ???????????? ???? = (???????????? ???????????? a. Column of A (n) and row of B (n) should always be the same in order to multiply them 3. Properties a. ???? ???? + ???? = ???????? + ???????? b. (???? + ???? ???? = ???????? + ???????? c. ???? ???????? = ???????????? d. In general, AB≠BA Suggested Homework: Study for quiz Section 7.1: 16, 18, 26 rd Math 340 Lab – Introduction to Ordinary Differential Equations – February 23 , 2016 What We Covered: 1. Quiz next class a. Highlights i. Covers sections 6.1 and 7.1 2. Course Content – Chapter 7: Matrix Algebra a. Section 7.1: Vectors and Matrices Continued i. Properties of Matrix Multiplied 1. ???? ???? + ???? = ???????? + ???????? 2. ???? ???????? = ???????? ???? ) 3. Identity 1 0 0 ???? = 0 1 0 0 0 1 4. ???????? = ???????? = ???? 0 −1 2 5 ???? = 1 −1 0 ???? = −1 −3 2 0 −2 −3 ???????? = 6 −17 ii.Example: 1. In general: ???? ???? = ???? ???? = ???? 1 ????2 ????3 ???? iii.A system would be… ???? ???????? ???????????????????????????????????????????? ????????= ???????????????????????????????? ????11 ???? ⋯ ????1???? ???? ???? 1 [ ⋮ ⋱ ⋮ ] ????????1 ???? ⋯ ???????????? ????= ???? ???? 1. It can be written as Ax=b which is matrix notation iv. Example: 3???? − ???? = 0 ???? + ???? = 1 ???????? = ???? ???? 3 −1 0∗ ???? = 0 0 1 1 1 ???? In ???????? = ????, ???? = 0 → ℎ???????????????????????????????????????? ???? ≠ 0 → ????????ℎ???????????????????????????????????????? v. Transpose 1. The transpose of a matrix with n rows and m columns will have m rows and n columns 2. ???? = transpose of A 0 1 −2 ???? 0 9 ???? = is the matrix ???? = 1 2 9 2 5 −2 5 3. Properties ???? ???? a. (???? ) = ???? b. (∝ ????) =∝ (???? ) b. Section 7.2: Systems of Linear Equations with Two or Three Variables i. Example: Find the solution set of 3???? + 4???? = 9 1. Matrix Notation ???? = 3 4 ???? = ???? ???? = 9 ???? ???? 3 4 ∗ ???? = 9 2. Parametric Form ???? 9 − 4???? 4???? = = 3 − ???? ???? ???? 3 3 4 = + − 3 0 ???? 3 −4 = + ???? 3 0 1 ???? = ???? + ???????? ???? ∈ ℝ 3. Augmented Matrix: ???????? = ???? (????|????) 1 −1 0 → ???? − ???? = 0 3???? + 4???? = 9 3 4 9 1 −1 0 → ???? − ???? = 0 7???? = 9 0 7 9 9 7???? = 9 → ???? = 7 Solving linear equations c. Section 7.3: Solving Systems of Equations i. Row Echelon Form 1. Definition: in each row that contains a pivot, the pivot lies to the right of the pivot in the preceding row. Any rows that contain only zeros must be at the bottom of the matrix ???? ∗ ∗ 0 ???? ∗ 0 0 ???? a. Essentially, every pivot is located to the right with respect to the previous one 2. Row Operations a. To get a matrix to look the way we want it to, we can manipulate each row with specific operations… i. R1: add a multiple of one row to a different row ii. R2: Interchange two rows iii. R3: Multiply a row by a nonzero number Suggested Homework: Section 7.1: 16, 18, 26 Section 7.2: 20, 22, 38 Section 7.3: 10, 22, 24 Study for quiz Math 340 Lecture – Introduction to Ordinary Differential Equations – February 24 , 2016 th What We Covered: 1. Quiz today! 2. Course Content – Chapter 7: Matrix Algebra a. Section 7.3: Solving Systems of Equations Continued i. Example: Find the solution ???? + 3???? + 2???? + 2???? = 1 2 3 4 5 ???? 1 2???? +23???? + 3???? + 74 = 8 5 2???? + 4???? + 6???? + 9???? + 15???? = 2 1 2 3 4 5 0 1 3 2 2 1 1 2 3 5 7 8 2 4 6 9 15 2 1. This is in Row Echelon Form a. Get a nonzero in the 11place ????1↔ ???? 2 1 2 3 5 7 8 0 1 3 2 2 1 2 4 6 9 15 2 b. Want a 130 −2???? 1 ???? 3 1 2 3 5 7 8 0 1 3 2 2 1 0 0 0 -1 1 -14 i. We have pivots in every column except the 3 rdand 5 , we can call these Free Columns or free variables which means they can be anything c. X3and x 5re free variable. x1, 2 , 4 depend on x3, 5 ????3= ???? ????5= ???? 3: −????4+ ???? 5 −14 −???? 4 ???? = −14 ???? = ???? + 14 4 2: ???? + 3???? + 2???? + 2???? = 1 2 3 4 5 ????2+ 3???? + 2 ???? + 14 + 2???? = 1 ???? = −27 − 3???? − 4???? 2 1: ???? + 2???? + 3???? + 5???? + 7???? = 8 1 2 3 4 5 ????1+ 2 −27 − 3???? − 4???? + 3???? + 5 ???? + 14 + 7???? = 8 ???? − 6???? + 3???? − 8???? + 5???? + 7???? = 8 1 ???? = 8 + 3???? − 4???? d. So the solution is… ????1= −8 + 3???? − 4???? ????2= −27 − 3???? − 4???? ????3= ???? ????4= 14 + ???? ???? 5 ???? -8 3 -4 -27 -3 -4 0 + t 1 +s 0 14 0 1 0 0 1 ii. Consistency 1. How do we know whether a system has any solutions at all? A system of equations is consistent if and only if the augmented matrix of an equivalent system in row echelon form has no pivot in the last column 2. Example 1 0 −2−1 0 2 4 2 0 0 0 −2 ????3: 0 ???? + 0 ???? + 0 ???? = −2( ) 0 = −2 a. No solution 3. Ax=b isn’t consistent if A can be reduced to row echelon form with no pivots in the last row Suggested Homework: Section 7.3: 10, 22, 24 th Math 340 Lecture – Introduction to Ordinary Differential Equations – February 26 , 2016 What We Covered: 1. Exam 1 Next Week! a. Highlights i. March 3 at 5pm in Wager 232 ii. Covers sections: 2.2, 2.3, 2.4, 2.6, 2.7, 2.9, 7.3, 7.4, 7.5, 7.6 iii. I will be uploading an original study guide on my site in Study Soup, and it only costs $8 iv. Good Luck! 2. Course Content – Chapter 7: Matrix Algebra a. Section 7.4: Homogeneous and Inhomogeneous Linear Systems i. Homogeneous Systems: a homogeneous system has the form Ax=0, where the right-hand side is the zero vector. Since A0=0, the homogeneous solution is always x=0 so therefore it’s consistent 1. It has a nontrivial solution if it can be reduced to a row echelon form with a free column and if it has fewer equations than unknowns ii. Example: −3 −2 4 ???? = 14 8 −18 4 −5 0 ???????????????????? ???????? = 0 1. Row reduce: 14 ???? + ???? → ???? , 4???? + ???? → ???? 3 1 2 2 3 1 3 3 −3 −2 4 4 2 0 − 3 3 2 1 0 − 3 3 2. 3????2→ ???? 2 3????3→ ???? 3 −3 −2 4 0 −4 2 0 −2 1 3. − ???? + ???? → ???? 2 2 3 3 −3 −2 4 0 −4 2 0 0 0 a. The last column is free because there’s no pivotZ = free variable, z = t Solution: ???? ???? 1 ???? = ????= ⁄ = ???? ⁄1 2 2 ???? ???? 1 Solution Set: 1 ???? {???? = ????(1,⁄2 ,1) |???? ∈ ℝ} = ???????????????? iii. Theorem: If p is a particular solution of Ax=b, then any solution is of the form x=p+tv, where Av=0 iv. Example: 0 1 3 2 2 1 ???????? ???? = 1 2 3 5 7 ???? = 8 2 4 6 9 14 2 ????ℎ???? ???????????????????????????????? ???????????? ???????? = ???? ???????? … ???? = -8 3 -4 -27 -3 -4 0 + s 1 + t 0 14 0 1 0 0 1 ???? = ???? + ????????1+ ????????2 ???? = ???? − ???? Is a solution ???????? = 0 ???? = ????????1+ ????????2 v. Nullspace 1. A nullspace of a matrix A is the set of all solutions to the homogeneous system of linear equations Ax=0. The nullspace of A is denoted by null(A) 2. Property: if x and y ∈ null (A) then a. ???? + ???? ∈ ???????????????? (????) b. ???????? ∈ ???????????????? ???? ,∀???? ∈ ℝ c. Solve ???????? = 0 ≡ ???????????????? ???????????????? (????) vi. Example: Using information from the previous example ( ) 1 ???? ???????????????? ???? = {????(1, ⁄2 ,1) :???? ∈ ℝ} ???????????????? ???? = {???????? +1???????? |2 1 ∈ ℝ} b. Section 7.5: Bases of a Subspace i. Definition: the span of1???? ,…???????? of ℝ span (1 ,…,???? ) = all linear combinations of ????1,…???? ???? ii.Example: Find the nullspace of C= 1 Cx=0 solve… 4 2 2 1 ???? 0 ∗ ???? = 4 2 0 2 1 0 − 2????1+ ???? 2 2 1 0 4 2 0 0 0 0 ???? = ???????????????? ???????????????????????????????? ???? = ???? ???? 2???? + ???? = 0 → 2???? = −???? → ???? = − 2 Solution- ???? ???? = ⁄2 = ????1⁄2 ???? ???? 1 ???????????????? ???? = {????( ,1) |???? ∈ ℝ} 2 1 ???? = ????????????????(( ⁄2 ,1) ) iii.Matrix Example (1 2 −2 |0) ???????????????????????????? ???????????? ???????????????????????????? ????ℎ???? ???????????????????? ???????????????????????????????????? ???????? ???? ????????????????????,????ℎ???? ????????ℎ???????????? ???????????? ???????????????? ???????????????????????????????????? ???????????? ???? = ???? ???????????? ???? = ???? ???? + 3???? − 2???? = 0 → ???? = 2???? − 3???? 1. Solution ???? 2???? − 3???? 2???? −3???? ???? = ???? = 0 + 1 ???? ???? 2 0 2 −3 = ????0 + ???? 1 1 0 ∗∗ ???????????????????????????? ????ℎ???????? ???????????????? ????ℎ???? ???????????????????????????????????? ∗∗ The nullspace of D= ???? ???? {????(2,0,1) + ????(−3,1,0) :????,???? ∈ ℝ} =????????????????((2,0,1) ,(−3,1,0) )???? Suggested Homework: Study for the midterm! Section 7.4: 4, 8, 10, 14, 18 Section 7.5: 5, 10, 20, 22, 30, 38

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