Errata for MIT OCW 18.06SC

This page has errata, cross-references, and other notes about the MIT Open Courseware subject 18.06SC Linear Algebra, led by Professor Gilbert Strang.

Contents

Textbook Cross-Reference

Index Errata for Differential Equations and Linear Algebra

Errata

"Overview of Key Ideas" Recitation

Problem 3.2

Matrix Names

Index for Differential Equations and Linear Algebra

Textbook Cross-Reference

The textbook for 18.06SC is Prof. Strang's Introduction to Linear Algebra from 2009 ("ila"). On the 18.06SC syllabus page is the suggestion that one might consider supplementing the course with Prof. Strang's 2014 textbook Differential Equations and Linear Algebra ("dela"). For those with a suitable background in mathematics (such as already knowing much of the material from having taken another linear algebra course in the past), or who are willing to augment the textbook with other resources (e.g. Wolfram MathWorld or Kahn Academy) might wish to get dela instead of ila, here is a table showing which parts of dela correspond most closely to the assigned readings in 18.06SC:

lecture	lecture title	reading assignment from Introduction to Linear Algebra	corresponding part(s) of Differential Equations and Linear Algebra
L01	The Geometry of Linear Equations	1.1, 1.2, and 2.1	4.1
L01-A	An Overview of Key Ideas	(note: see erratum concerning the recitation)
L02	Elimination with Matrices	2.2, 2.3	4.2
L03	Multiplication and Inverse Matrices	2.4, 2.5	4.3, 4.4
L04	Factorization into A = LU	2.6	Ch. 4 notes on p. 249 and appendix on p. 490
L05	Transposes, Permutations, Spaces ℝⁿ	2.7	4.5
L06	Column Space and Nullspace	3.1, 3.2	5.1, 5.2
L07	Solving Ax = 0: Pivot Variables, Special Solutions	3.2	5.2
L08	Solving Ax = b: Row Reduced Form R	3.3, 3.4	?, 5.3
L09	Independence, Basis, and Dimension	3.5	5.4
L10	The Four Fundamental Subspaces	3.6	5.5
L11	Matrix Spaces; Rank 1; Small World Graphs	3.3, 8.2	5.5, 5.6
L12	Graphs, Networks, Incidence Matrices	8.2	5.6, 7.5
L13	Quiz 1 Review	Chapters 1, 2, and 3; 8.2	Chapters 4 and 5; 8.2
lecture	lecture title	reading assignment from Introduction to Linear Algebra	corresponding part(s) of Differential Equations and Linear Algebra
L14	Orthogonal Vectors and Subspaces	4.1	5.5?
L15	Projections onto Subspaces	4.2	7.1
L16	Projection Matrices and Least Squares	4.3	7.1
L17	Orthogonal Matrices and Gram-Schmidt	4.4
L18	Properties of Determinants	5.1	? (but see appendix on p. 492)
L19	Determinant Formulas and Cofactors	5.2
L20	Cramer's Rule, Inverse Matrix, and Volume	5.3	?
L21	Eigenvalues and Eigenvectors	6.1, 6.2	6.1
L22	Diagonalization and Powers of A	6.2	6.2
L23	Differential Equations and exp(At)	6.3	6.3
L24	Markov Matrices; Fourier Series	8.3, 8.5
L24b	Quiz 2 Review	Chapters 4 and 5; 6.1-6.3, 8.3, 8.5
lecture	lecture title	reading assignment from Introduction to Linear Algebra	corresponding part(s) of Differential Equations and Linear Algebra
L25	Symmetric Matrices and Positive Definiteness	6.4, 6.5	6.5, 7.2
L26	Complex Matrices; Fast Fourier Transform	10.2, 10.3	?, 8.2
L27	Positive Definite Matrices and Minima	6.5	7.2
L28	Similar Matrices and Jordan Form	6.6	7.2
L29	Singular Value Decomposition	6.7	7.2
L30	Linear Transformations and Their Matrices	7.1
L31	Change of Basis; Image Compression	7.2
L32	Left and Right Inverses; Pseudoinverse	7.3
L33	Quiz 3 Review	Chapters 6 and 7; 10.2, 10.3

Index Errata for Differential Equations and Linear Algebra

A complete corrected index is here.

This is a list of headings for which I have found specific errors and also managed to work out the proper page numbers. Early in my use of DE&LA I found many cases, not listed here, for which I could not figure out the correct page number. However it seems that the errors are methodical: a page number in the high 100's or low 200's is off by one, a page number in the low 300's is off by two and in the mid 300's to low 400's is off by three. By looking at these corrected index entries you can see the pattern in more detail:

column space 252, 257, 276 254, 259, 278

conductance matrix 124, 382, 423, 424 385, 425, 426

determinant 174, 227, 231, 323, 327, 329, 333, 344, 350, 399, 228, 232, 326, 330, 332, 336, 347, 353, 402, 492

Fast Fourier Transform 88, 449 450-452

free-free boundary conditions 409 412

Fundamental Theorem 5, 8, 42, 243, 302, 305, 397 244, 304, 307, 400

heat equation 407 410, 455, 456

Laplace equation 414, 415 416, 417

Laplace transform, 121, 140-150151, 470-478

Laplace's equation 416, 440, 441 418, 442, 443

perpendicular subspaces 310 306

rank one matrix 303 305

second difference 239, 245, 407, 412, 413 240, 246, 410, 415, 416

shift invariance 98, 459 461, 480, 482, 487

solution curve 153 154

trace 174, 328, 329, 333, 344, 350, 381 175, 331, 332, 336, 347, 353, 384

Wronskian 134, 135, 363, 381 366, 384

When searching for the proper index entry, do not skip over the Problem Sets, where many concepts are introduced and/or expanded upon.

{Speculation: perhaps the Chapter 3 intro. on page 153, Chapter 4 Notes on p. 249 and Chapter 5 Notes on p. 321, or the blank pages immediately preceding/following them, were inserted after the index was completed. Also, something seems to have been removed between sections 8.2 (on FFT) and 8.3 on (Heat Equation).}

ERRATA

Following are uncorrected errors that I found whilst going through 18.06SC. I do not mention transient mistakes, such as a number written on the blackboard which is soon fixed (often after being pointed out by a student who was in the classroom at the time).

"Overview of Key Ideas" Recitation

After the first lecture you will come to the segment "An Overview of Key Ideas". The lecture (derived from the beginning of subject 18.085) fits well at this point in the course, but the recitation (problem solving video) does not. If you wish to do the recitation anyway, you might wish to review the textbook reading for lecture L08, particularly section 3.4 "The Complete Solution to Ax = b" (or section 5.3 in Differential Equations and Linear Algebra). Since there is no problem set at this segment, you might just save the recitation until you get to L08.

Problem 3.2

In the answer for problem 3.2, the first mention of "cR₂" has the wrong subscript; it should instead be cR₃.

Lecture 6 summary

In the "Nullspace of A" section, where it says "A(x₁+x₂) = Ax₁ + x₂ = 0+0", there is an A missing; it should say "A(x₁+x₂) = Ax₁ + Ax₂ = 0+0".

Problem 7.1

In the answer for problem 7.1, there are two sign errors: the final step of row-reduction should give ²³/₄ in row 1, column 3, and so the corresponding special solution in the answer to part c) should have -²³/₄ in its first component.

Recitation 12

The third part of recitation 12 asks for the "trace" of a matrix. This term is not mentioned in the lecture, but is in the reading using the recommended textbook Introduction to Linear Algebra (Strang, 2009).

If you are instead using Differential Equations and Linear Algebra (Strang, 2014) you'll find that the index entres for "trace" are wrong. See my index errata, or just read the discussion of determinants at the end of section 6.1 (pages 331, 332).

Matrix Names

A non-specific name for a matrix

A^-1 the inverse of A

A^T the transposition (or "transpose") of A

C circulant

D difference matrix

E elimination matrix

F Fourier matrix

I the identity matrix

L lower triangular matrix

P permutation matrix

Q orthogonal matrix

S symmetric matrix
second-order difference matrix

U upper triangular matrix

Index for Differential Equations and Linear Algebra

The index in the book Differential Equations and Linear Algebra is riddled with errors. (It seems they indexed a draft version and in the final, all the page numbers changed, in a fairly monotinic but non-linear manner.) See index errata for a few corrections.

I decided to re-index the whole book in parts, as part of my exam review and test prep process. Here then is a partial index

Sections indexed so far: 4.1-4.4

back substitution 249 block multiplication 227 coefficiant matrix 199 column picture 198, 204, 206 column vectors, combination 198, 201 complete solution 203, 205 dependent vectors 205 determinant 228 2x2 case 1/(ad-bc) 228 and invertibility 232 as product of pivots 232 difference matrix 240, 246 dot product (of vectors) 201 of perpendicular vectors 201 eigenvalue of symmetric matrix 239 eigenvector of symmetric matrix 239 elimination 210, 215-218 see also Gauss-Jordan elimination as product of D^-1, E, and P 232 as proof of invertibility 233 back-substitution 213, 214 complete solution 211 infinitely many solutions 211 no solution 211 pivots on diagonal 213 row exchange 212 singular system 211 upper triangular result 213 elimination matrix 221, 222 factorisation as A=LU 249 as A=SQ 244 forward substitution 204, 249 Gauss-Jordan elimination 230-232, 236 and U L^-1 232 of triangular matrix 233 Hadamard matrix 243 identity matrix 201 independent vectors 205 inverse matrix 220, 221, 228 by Gauss-Jordan elimination 230-232 existence 228 number of operations n³ 232 of 2x2 matrix 228 of diagonal matrix 229 of elimination matrix 229 of product (AB)^-1 = A^-1 B^-1 229, 230 of a sparse matrix 232 of square matrix 230 transpose of 238 uniqueness 228 invertible matrix 204, 205, 228 and Av=0 228 and determinant 228 and uniqueness of Av=b 228 vs singular 232 LAPACK 242 least squares solution 239 linear combination of columns 199 linear combination of vectors 200, 201 magic squares 209 matrix associative property for addition A+(B+C)=(A+B)+C 220 circulant 205 commutative property for addition A+B=B+A 220 distributive property for scalar multiplcation c(A+B)=cA+cB 220 invertible 204 lower triangular 204 multiplication see matrix multiplication powers of 221, 225, 226 singular 202, 205 matrix multiplication A_mxn B_nxp 219, 223, 224, 249 AB != BA 219, 220 1. as mxp dot products (aggregate of row-column products) 222, 226 2. as p matrix-vector products 219, 222, 226 3. as m vector-matrix products 222, 226 4. as 1 matrix-sum of n column-row products 222, 226 associative property (AB)C=A(BC)=ABC 220 blocks see block multiplication by identity matrix 219, 221 by inverse of a matrix 220 distributive property A(B+C)=AB+AC 220 distributive property (A+B)C=AC+BC 220 number of operations mxnxp or n³ 223, 227 matrix-vector multiplication 199, 202, 207, 208 as dot products of rows 202 as a combination of columns 202 multiplier (of a row) 210 null solution 203, 205, 211 orthogonal matrix 238, 242, 247 columns of 238 (left) inverse of see orthogonal matrix, transpose of product of 244 symmetric 244 transpose of 242, 243 and unit vectors 238 orthonormal see orthogonal particular solution 203, 205, 211 permutation matrix 241, 246 to select largest pivots 242 perpendicular vectors 201 see also orthogonal pivots 210, 211 and invertibility 233 rotation matrix 238 row picture 197, 204, 206 scalar multiplication 198 second difference matrix 240, 246 compared to L=AA^T 241 singular matrix 202, 205 no solution or infinity of solutions 203 sudoku 209 sum matrix 249 symmetric matrix 238, 245 A^TA 239 eigenvalue of 239 eigenvectors of 239 and least squares solution 239 orthogonal 244 transpose (of a matrix) 238, 245 inverse of 238 products of 238 tridiagonal matrix 232, 246 triangular matrix invertibility 233 unit vector 238 see also orthonormal upper triangular 210 vector addition 199, 200

References

Strang, Gilbert. 18.06SC Linear Algebra, Fall 2011. (MIT OpenCourseWare: Massachusetts Institute of Technology), http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011 (Accessed 17 Dec, 2014). License: Creative Commons BY-NC-SA

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