MATH - Fall, 5 Beer is proof that God loves us and wants us to be happy. Benjamin Franklin { } Errata (fancy word for screw ups) for chapter and. The notation q, p { } down from the top of the page (correction) means on page q on the p th line there is up from the bottom of the page this (correction). 5 after the third equal replace u with u (twice) 8 v in (.7) should be v 8 + ± should be + 9 4 = (missing minus) 9 3 (.) should be (.5) 3 4, 5 Replace the word vertexes with vertices. See below. 3 Figure. Compare below with your text 34 6 parallel. (missing period) 34 Figure.4 see the next page 4 7. remove period 37 3 (.8) (in the matrix) should be a 33 4 7. remove period 4 5 Replace (.8) with (.9). 4 8 Replace (.4) with (.7). 4 9 Replace x + 3x + = 8 + 3( 4) + = 3 with x + 3x + = 8 + 3( 4) + =. 46 5 Replace the word right with left. 48 3 (.36) should be (.9) 49 3 (.36) should be (.9) 49 (.37) b 3 should be 5 should be 57 3 Replace with in x p 57 6 Replace Let with Letting 57 5 Replace / with / 57 3 Replace x + x = with x + (/)x = 57 Replace the [ ] in the last row of A with [ / ] 53 4 b = [ 3 ] T b = is missing 54 6 If x = then x = / 7 and finally... / should be /
. Planes and Normals The geometry used in Figure. to illustrate the solutions (or lack of solutions) of equations in two unknowns can be extended to equations in three unknowns. A portion of the graph (the triangular region) of the plane defined by the equation x + x + 3 = N x = 3 x x = 6 () [ ] T N = [ 3] T x + x + 3 = 6 [ 3 ] T x x [6 ] T Figure. is shown in Figure.. Notice for example, if x =, then the graph of the equation reads 6 x + 3 = 6 which is the straight line passing through the two vertices and. x + x + 3 = 6 line of intersection x N x + 5x + 4 = 4 N x Figure.4 The equation of the lines through the other two legs of the triangle are obtained by setting
3 x = (x + 3 = 6) and = (x + x = 6), respectively. The vector N = called a normal to the plane. It is orthogonal to every vector in the plane. This will be shown for arbitrary planes. Here are a few more revelations from Thirsty Bud. 69 The resulting matrix is m p. (It should read m n. The second m p in the line.) 69 6 In part, (The comma is missing.) 69 7 The resulting matrix is m p. (It should read m n.) 7 7 The a 3 element in the matrix product A 3 A 4 should read () + 3() + () = 6. 7 3 [B T A T ] nm, where the product (This is in part 3 of Theorem 3.7. The word where is missing.) 75 5 Replace b by 3 (twice) in the displayed equation (3.8) 77 the matrix A.. (Delete one of the two periods). [ ] αa αa 85 7 write B = delete the αa = a a This is in Example 3. in the first displayed line. 9 8 In the matrix A change a = 3 to a = 5 and the last row from 3 4 5 to 3. 93 will also give (The word give is missing). 93 6 The displayed line should read q = p p = [ ] and q = p p = [ 94 respectively. (The period is missing). 95 4 The (3.8) should be (3.6). In Exercise 3.3 part 3. 95 5 Definition(3.) should be Definition(3.4). In the first line of Exercise 3.5 99 is the inverse of U (The of is missing). 5 Definition(.ed) should be Definition(.) 6 it s should be its 56 7 Parts,, and 3 should read Parts,,3 and 4 56 4 replace 4 by in Part 3., replace by in Part 4. 4 58 The 3, element of L should be (not a 64 4/6 The, and, elements of DC should be 5 and 9, respectively. not and 7. Same change in C T D T. 3 ] is
4 4 hours in a day, 4 beers in a case. Coincidence, I think not... Stephen Wright The work of old Thirsty Bud continues. He doesn t tell me anything anymore. 4 8 satisfy following (It should read: satisfy the following 5 3 v, v, is unique. (The comma is missing.) 5 In words, V a (The comma is missing.) 6 Leave out the first sentence of Defnition 4.4. Replace it with: Let S = { v, v,..., v m } denote a set of m elements in a vector space V. Now continue: The vector u =.... 9 8 span{ a, a } should read span{ a, a 3 } 4 span{ a, a } should read span{ a, a } 6 Occurs in displayed equation (4.). Replace in C A the vector [ 3 ] T with [ ] T and [ 3 4 ] T with [ 5 3 ] T and in C the vector [ ] T with [ ] T. 4 3 Replace R 3 = N A NA = N A RA = RA R A should read R 3 = N (A) N (A) = N (A) R A = RA R A or more simply R 3 = N (A) N (A) = RA R A 4 7/8 Replace R 4 = N A NA = N A RA = RA R A should read R 4 = N (A) N (A) = N (A) R A = RA R A or more simply R 4 = N (A) N (A) = RA R A 4 v V should read v V 46 6 N A should read N (A). Two times in this line. 5 7 Need a period at the end of the displayed equation. 5 parameter in A (the in is missing). 54 3 N A should be N (A) 56 8 There should be a comma between w and w. 58 6 Replace parts,, and 3. by parts and. 58 Replace part 3 by part. 64 4 A = should read A x = 66 6 a 33 = should read a 33 =. 68 impliesδ should read implies δ 7 4 Need a period at the end of the displayed equation. 7 Part should read In Part, 7 N (B T b) should read N (B T B) 7 4 In this case it is also the solution which... should read In this case x ls is also the traditional solution which... 73 4 normal equations reads should read normal equations [ ] 3 5 75 7 A =.The period is missing 6 76 6 ( + b b ) 3 + b = 7 + 5b b.the should be =.
5 A woman drove me to drink, and I hadn t even the courtesy to thank her. W.C. Fields 75 4 A p j = λ j p j, j =, The comma is missing. 94 Displayed equation (5.8): with inverse Q... instead of: has the inverse 5 8 to closely should be too closely. 3 it s should be its too closely. 5 3 Theorem(5.) should be Theorem(5.) 6 and a be real... the word be is missing. 6 3 delete the second A in equation(5.45) 6 6 a > not a > 7 The first a 33 should be a [ ] [ ] 3 b b 3 5 P = ( u u T ), replace a with a. a a 3 Need a period at the end of part. 88 7 p p = missing = sign Thirsty Bud s work is now done. 8 the second p p should be p p An amazing formula attributed to Euler is Math made simple e πi + = which connects everybodies favorite constants: e is the Naperian base for logarithms, π is the area of a circle of radius, is the additive identity, is the multiplicative identity and i = is the complex unit. Actually, and this really torques me off, the number didn t make the formula and, to this day, I m still pissed off at Euler. After all, the number is the only even prime. It makes one think that maybe Euler was prejudiced. Prejudice is not just a st (there s that number again) century phenomena. I mean Euler could have written e πi + = or ( e πi + ) = which, and this is a st century phenomena, is more inclusive. It s all just so depressing.