# 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

Errata

### 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

AT 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

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2019 Jan 05. s.11