201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes Tristan Martin Fall 2017 Enriched 201-NYC-05 (Enriched Linear Algebra I), instructed by Matthew Egan, Yariv Barsheshat, Christopher Turner, and Dominic Lemelin. These notes probably contain many typos. Contents 1 Introduction to Linear Systems and How to Solve Them (Egan)4 1.1 Linear equations and linear systems ..................... 4 1.2 Introduction to augmented matrices..................... 5 2 Applications of Manipulations that Leave the Solution Set Unchanged (Egan)6 3 More on Row Reduction (Egan)9 3.1 REF and RREF ................................ 9 3.2 Row vectors, column vectors and linear dependance ............ 10 3.3 Matrix rank and linear systems........................ 10 4 Applications of Row Reduction and Introduction to Homogenous Systems (Egan) 13 4.1 Some applications ............................... 13 4.1.1 Network flow............................... 13 4.1.2 Balancing chemical reactions...................... 13 4.2 Homogenous systems and solutions to linear systems............ 14 5 Introduction to Geometric Vectors (Lemelin) 16 5.1 Vectors: the geometric case.......................... 16 6 Notation (Turner) 17 7 Dot Product, Projections, Orthogonals and the Cross Product (Barsheshat) 18 7.1 Dot product .................................. 18 7.2 Projection of a vector onto another vector.................. 20 7.3 Orthogonal projection............................. 20 7.4 Cross product.................................. 21 8 Lines and Planes (Barsheshat) 23 2 8.1 Lines in R ................................... 23 3 8.2 Lines in R ................................... 23 8.3 Parallel lines .................................. 24 1 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes 3 8.4 Planes in R .................................. 25 8.4.1 Parallel planes.............................. 27 9 Span and Linear Combination, Dependence and Independence (Barsheshat) 29 10 Span and Bases (Barsheshat) 31 10.1 More on span and linear combination, dependence and independence . 31 m 10.2 Basis for R .................................. 32 11 Subspaces and Span (Barsheshat) 34 12 Basis and Dimension for a Subspace (Barsheshat) 36 12.1 Basis for a subspace .............................. 36 12.2 Dimension of a subspace............................ 36 13 Exercises on Span (Barsheshat) 38 14 Subspaces of Matrices (Barsheshat) 40 15 Matrix Operations (Barsheshat) 42 15.1 Matrix addition and scalar multiplication.................. 42 15.2 Matrix multiplication ............................. 42 16 Column and Row Representations of Matrix Multiplication and the Prop- erties of the Operation (Barsheshat) 44 16.1 Partitioning matrices.............................. 44 16.2 Column representation of matrix multiplication............... 44 16.3 Row representation of matrix multiplication................. 44 16.4 Column-row representation .......................... 44 16.5 Algebraic proterties of matrix addition and scalar multiplication . 45 16.6 Matrix exponentiation............................. 46 16.7 Transpose of matrix .............................. 46 16.8 Properties of matrix multiplication...................... 46 17 Linear Systems With Matrices (Barsheshat) 47 18 Matrix Inverses (Barsheshat) 49 18.1 Using row-reduction to find inverses ..................... 50 19 Gauss-Jordian Algorithm for Finding Inverses (Barsheshat) 51 20 More on Matrix Inverses 54 21 Determinants (Barsheshat) 55 21.1 Determinants for 3 × 3 matrices ....................... 56 22 More on Determinants (Barsheshat) 57 22.1 Generalized cofactor expansions ....................... 57 23 Even More On Determinants (Barsheshat) 59 23.1 Triangular Matrices .............................. 59 23.2 Determinants of elements matrices...................... 59 23.3 Cramer's Rule ................................. 60 2 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes 24 Complex Numbers (Barsheshat) 62 24.1 Operations on complex numbers ....................... 62 24.2 Polar form of copmlex numbers........................ 63 25 More on Complex Numbers (Barsheshat) 65 25.1 De Moivre's formula.............................. 65 25.2 Basic polynomial of complex numbers.................... 66 26 Integrative Activity: Electrical Circuits (Barsheshat) 67 27 Enriched Material: Eigenvalues and Eigenvectors (Barsheshat) 68 27.1 Finding eigenvalues of A ............................ 68 3 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes §1 Introduction to Linear Systems and How to Solve Them (Egan) §1.1 Linear equations and linear systems The main field of study for this course will be linear equation which have the form a1x1 + a2x2 + a3x3 + ··· + anxn = b; (1) a1; a2; a3; ··· ; an; b 2 R and x1; ; xn are unknowns. A solution of a linear sysrem is a set of values fs1; ··· ; sng. that make the equation true when we replace fx1; ··· ; xng with fs1; ··· ; sng. Definition 1.1. A linear system of linear equations is a collection of m linear equations in n unknowns. It thereby has the form 8 a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1 > > a21x2 + a22x2 + a23x3 + ··· + a2nxn = b2 <> a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3 (2) > . > . > :am1x1 + am2x2 + am3x3 + ··· + amnxn = bm Definition 1.2. A solution to a linear system is a set of values fs1; ··· ; sng that is a solution for all equations in the system. Example 1.3 Consider the 2 × 2 system x − 2x = −1 1 2 (3) −x1 + 3x2 = 3 It has unique solution (3; 2) and the system is therefore consistent. Example 1.4 Consider the 2 × 2 system x − 2x = −1 1 2 (4) −x1 + 2x2 = 3 It has no solution (i.e., the system is inconsistent). Example 1.5 Consider the 2 × 2 system x − 2x = −1 1 2 (5) −x1 + 2x2 = 1 It has infinitely many solutions and is hence a consistent system. 4 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes Theorem 1.6 Every linear system of equations has either 0,1 or 1 solutions. Furthermore, to describe the complete solution set for example5, we use parameters to write a out general solution. If we define x2 = t 2 R, we can write x1 as x1 = 2x1 − 1 (6) = 2t − 1: (7) The former is called the parametric eqautions of a line. The general solution is written as x = 2t − 1 1 (8) x2 = t Choose a value for t to obtain a particular solution. For instance t = 0 gives solution (−1; 0) and t = 1 gives (1; 1). §1.2 Introduction to augmented matrices Now, consider larger linear systems like the following 3 × 3 system: 8 < x1 − 2x2 + x3 = 0 2x2 − 8x3 = 8 (9) :−4x1 + 5x2 + 9x3 = −9 To solve such a system we first need to strip away all but the coefficients and the constant to obtain the augmented matrix of the system: 0 1 −2 1 0 1 @ 0 2 −8 8 A : (10) −4 5 9 −9 Remark 1.7. What manipulations of the system will leave the solution set unchanged? We can: • interchange any two rows (equations) • multiply a row (equation) by a non-zero constant • replace a row (equation) by itself plus a multiple of another row (equation) 5 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes §2 Applications of Manipulations that Leave the Solution Set Unchanged (Egan) Continuing our example from lecture 1, we will use these properties as a means to solve the system. We will soon learn there is an algorithm (i.e., Gauss-Jordan elimination) that allows us to apply these transformations to any augmented matrix to obtain a solution set to the original system of equations. For now, we will gloss over the definition of row echelon form (REF) and reduced row echelon form (RREF), but from the upcoming examples, it will be pretty clear. Now, back to our augmented matrix from lecture 1. For now, just focus on under- standing how the discussed manipulations are applied (noting that Ri indicates the ith row), the desired result will be later studied. 0 1 −2 1 0 1 01 −2 1 0 1 R3+4R1 @ 0 2 −8 8 A −−−−−! @0 2 −8 8 A (11) −4 5 9 −9 0 −3 13 −9 0 1 0 1 1 −2 1 0 1 1 −2 1 0 2 R2 @0 2 −8 8 A −−−! @0 1 −4 4 A (12) 0 −3 13 −9 0 −3 13 −9 01 −2 1 0 1 01 −2 1 01 R3−3R2 @0 1 −4 4 A −−−−−! @0 1 −4 4A (13) 0 −3 13 −9 0 0 1 3 01 −2 1 01 01 −2 0 −31 R2+4R3 @0 1 −4 4A −−−−−! @0 1 0 16 A (14) 0 0 1 3 R1−R3 0 0 1 3 01 −2 0 −31 01 0 0 291 R1+2R2 @0 1 0 16 A −−−−−! @0 1 0 16A (15) 0 0 1 3 0 0 1 3 This matrix is in RREF. Note that these manipulations cannot be done simultaneously, R2+4R3 but one after the other. Hence the notation −−−−−! indicates that the operation R2 +4R3 R1−R3 was first conduction, then followed by R1 − R3. From this matrix, the systems solution is obvious, whereas with our augmented matrix, the solution was not. We therefore find: 8 <1 · x1 + 0 · x2 + 0 · x3 = x1 = 29 0 · x1 + 1 · x2 + 0 · x3 = x2 = 16 (16) : 0 · x1 + 0 · x2 + 1 · x3 = x3 = 3 Note that the 1 are called pivot entries or leading coefficients. Technically, a pivot entry does not necessarily need to be 1, but depending on the textbook you use, some authors are quite pernickety about solely using 1s as pivot entries. Hence, to accommodate the most textbooks possible, I will only be using 1s as pivot entries in these notes, but be aware that it is not required. We can write our solution in another form using column vectors (objects we will study later on in the course): 0 1 0 1 x1 29 @x2A = @16A : (17) x3 3 6 Tristan Martin (Fall 2017) 201-NYC-05-E (Enriched Linear Algebra I) Lecture Notes Note that we can verify our answer by plugging in our solution into the original. This is quite trivial, however, so I will leave it for you to do as an exercise. Exercise 2.1. Find the augmented matrix of the following system and try to row reducing it: 8 < x2 − 4x3 = 8 2x1 − 3x2 + 2x3 = 1 (18) :5x1 − 8x2 + 7x3 = 1 We therefore have the following augmented matrix which can be reduced in the following way: 00 1 −4 81 02 −3 2 11 R1$R2 @2 −3 2 1A −−−−−! @0 1 −4 8A (19) 5 −8 7 1 5 −8 7 1 0 1 0 1 2 −3 2 1 5 2 −3 2 1 R3− 2 R1 @0 1 −4 8A −−−−−! @0 1 −4 8 A (20) 1 3 5 −8 7 1 0 − 2 2 − 2 0 1 0 1 2 −3 2 1 1 2 −3 2 1 R3+ 2 R2 @0 1 −4 8 A −−−−−! @0 1 −4 8 A (21) 1 3 5 0 − 2 2 − 2 0 0 0 − 2 We stop here.