 4.4.1: An integer is even if, and only if, _____.
 4.4.2: An integer is odd if, and only if, _____.
 4.4.3: An integer n is prime if, and only if, _____.
 4.4.4: The most common way to disprove a universal statement is to nd _____.
 4.4.5: According to the method of generalizing from the generic particular...
 4.4.6: Tousethemethodofdirectprooftoproveastatementofthe form,Forall x ina...
 4.4.7: Use the denitions of even, odd, prime, and composite to justify eac...
 4.4.8: Use the denitions of even, odd, prime, and composite to justify eac...
 4.4.9: Use the denitions of even, odd, prime, and composite to justify eac...
 4.4.10: Prove the statements . There are integers m and n such that m > 1 a...
 4.4.11: Prove the statements. There are distinct integers m and n such that...
 4.4.12: Prove the statements. There are real numbers a and b such that a+b ...
 4.4.13: Prove the statements. There is an integer n > 5 such that 2 n 1 is ...
 4.4.14: Prove the statements. There is a real number x such that x > 1 and ...
 4.4.15: Prove the statements . Thereisaperfectsquarethatcanbewrittenasasumo...
 4.4.16: Prove the statements . There is an integer n such that 2n2 5n+2 is ...
 4.4.17: Disprove the statements by giving a counterexample. For all real nu...
 4.4.18: Disprove the statements by giving a counterexample. For all integer...
 4.4.19: Disprove the statements by giving a counterexample. For all integer...
 4.4.20: Determinewhetherthepropertyistrueforallintegers, true for no intege...
 4.4.21: Determinewhetherthepropertyistrueforallintegers, true for no intege...
 4.4.22: Determinewhetherthepropertyistrueforallintegers, true for no intege...
 4.4.23: Prove the statements by the method of exhaustion. Every positive ev...
 4.4.24: Prove the statements by the method of exhaustion.Foreachintegern wi...
 4.4.25: a. Rewritethefollowingtheoreminthreedifferentways:as , if _____ the...
 4.4.26: (a) rewrite the statement with the quantication implicit as If ____...
 4.4.27: (a) rewrite the statement with the quantication implicit as If ____...
 4.4.28: (a) rewrite the statement with the quantication implicit as If ____...
 4.4.29: (a) rewrite the statement with the quantication implicit as If ____...
 4.4.30: Prove the statements. In each case use only the denitions of the te...
 4.4.31: Prove the statements. In each case use only the denitions of the te...
 4.4.32: Prove the statements. In each case use only the denitions of the te...
 4.4.33: Prove the statements. In each case use only the denitions of the te...
 4.4.34: Prove the statements. In each case use only the denitions of the te...
 4.4.35: Prove the statements. In each case use only the denitions of the te...
 4.4.36: Prove the statements. In each case use only the denitions of the te...
 4.4.37: Prove the statements. In each case use only the denitions of the te...
 4.4.38: Prove the statements. In each case use only the denitions of the te...
 4.4.39: Prove the statements. In each case use only the denitions of the te...
 4.4.40: Prove the statements. In each case use only the denitions of the te...
 4.4.41: Prove that the statements are false. There exists an integer m 3 su...
 4.4.42: Prove that the statements are false. There exists an integer n such...
 4.4.43: Prove that the statements are false. There exists an integer k 4 su...
 4.4.44: Find the mistakes in the proofs.Theorem: For all integers k, ifk > ...
 4.4.45: Find the mistakes in the proofs. Theorem: The difference between an...
 4.4.46: Find the mistakes in the proofs.Theorem: For all integers k, ifk > ...
 4.4.47: Find the mistakes in the proofs. Theorem: The product of an even in...
 4.4.48: Find the mistakes in the proofs. Theorem: The sum of any two even i...
 4.4.49: Determine whether the statement is true or false. Justify your answ...
 4.4.50: Determine whether the statement is true or false. Justify your answ...
 4.4.51: Determine whether the statement is true or false. Justify your answ...
 4.4.52: Determine whether the statement is true or false. Justify your answ...
 4.4.53: Determine whether the statement is true or false. Justify your answ...
 4.4.54: Determine whether the statement is true or false. Justify your answ...
 4.4.55: Determine whether the statement is true or false. Justify your answ...
 4.4.56: Determine whether the statement is true or false. Justify your answ...
 4.4.57: Determine whether the statement is true or false. Justify your answ...
 4.4.58: Determine whether the statement is true or false. Justify your answ...
 4.4.59: Determine whether the statement is true or false. Justify your answ...
 4.4.60: Determine whether the statement is true or false. Justify your answ...
 4.4.61: Determine whether the statement is true or false. Justify your answ...
 4.4.62: Determine whether the statement is true or false. Justify your answ...
 4.4.63: Determine whether the statement is true or false. Justify your answ...
 4.4.64: Determine whether the statement is true or false. Justify your answ...
 4.4.65: Determine whether the statement is true or false. Justify your answ...
 4.4.66: Determine whether the statement is true or false. Justify your answ...
 4.4.67: Suppose that integers m and n are perfect squares. Then m + n + 2mn...
 4.4.68: If p is a prime number, must 2p 1 also be prime? Prove or give a co...
 4.4.69: If n is a nonnegative integer, must 22n +1 be prime? Prove or give ...
 4.4.70: To show that a real number is rational, we must show that we can wr...
 4.4.71: An irrational number is a _____ that is _____.
 4.4.72: Zero is a rational number because _____.
 4.4.73: The numbers are all rational. Write each number as a ratio of two i...
 4.4.74: The numbers are all rational. Write each number as a ratio of two i...
 4.4.75: The numbers are all rational. Write each number as a ratio of two i...
 4.4.76: The numbers are all rational. Write each number as a ratio of two i...
 4.4.77: The numbers are all rational. Write each number as a ratio of two i...
 4.4.78: The numbers are all rational. Write each number as a ratio of two i...
 4.4.79: The numbers are all rational. Write each number as a ratio of two i...
 4.4.80: The zero product property, says that if a product of two real numbe...
 4.4.81: Assume that a and b are both integers and that a =0 and b =0. Expla...
 4.4.82: Assume that m and n are both integers and that n =0. Explain why (5...
 4.4.83: Prove that every integer is a rational number.
 4.4.84: Fill in the blanks in the following proof that the square of any ra...
 4.4.85: Consider the statement: The negative of any rational number is rati...
 4.4.86: Consider the statement: The cube of any rational number is a ration...
 4.4.87: Determine which of the statements are true and which are false. Pro...
 4.4.88: Determine which of the statements are true and which are false. Pro...
 4.4.89: Determine which of the statements are true and which are false. Pro...
 4.4.90: Determine which of the statements are true and which are false. Pro...
 4.4.91: Determine which of the statements are true and which are false. Pro...
 4.4.92: Determine which of the statements are true and which are false. Pro...
 4.4.93: Use the properties of even and odd integers . Indicate which proper...
 4.4.94: Use the properties of even and odd integers . Indicate which proper...
 4.4.95: Use the properties of even and odd integers . Indicate which proper...
 4.4.96: DerivethestatementsascorollariesofTheorems4.2.1, 4.2.2, and the res...
 4.4.97: DerivethestatementsascorollariesofTheorems4.2.1, 4.2.2, and the res...
 4.4.98: DerivethestatementsascorollariesofTheorems4.2.1, 4.2.2, and the res...
 4.4.99: It is a fact that if n is any nonnegative integer, then1+1 2 +1 22 ...
 4.4.100: Supposea,b,c,andd areintegersanda =c.Supposealso that x is a real n...
 4.4.101: Supposea, b, andc are integers and x, y, andz are nonzero real numb...
 4.4.102: Prove that if one solution for a quadratic equation of the form x2 ...
 4.4.103: Provethatifarealnumberc satisesapolynomialequation of the form r3x3...
 4.4.104: Prove that for all real numbers c, ifc is a root of a polynomial wi...
 4.4.105: When expressions of the form (x r)(x s) are multiplied out, a quadr...
 4.4.106: Observe that (x r)(x s)(x t) = x3 (r +s+t)x2 +(rs+rt+st)x rst. a. D...
 4.4.107: nd the mistakes in the proofs that the sum of any two rational numb...
 4.4.108: nd the mistakes in the proofs that the sum of any two rational numb...
 4.4.109: nd the mistakes in the proofs that the sum of any two rational numb...
 4.4.110: nd the mistakes in the proofs that the sum of any two rational numb...
 4.4.111: nd the mistakes in the proofs that the sum of any two rational numb...
 4.4.112: To show that a nonzero integer d divides an integer n, we must show...
 4.4.113: To say that d divides n means the same as saying that _____ is divi...
 4.4.114: If a and b are positive integers and ab, then _____ is less than o...
 4.4.115: For all integers n and d, d n if, and only if, _____.
 4.4.116: If a and b are integers, the notation ab denotes _____ and the not...
 4.4.117: The transitivity of divisibility theorem says that for all integers...
 4.4.118: The divisibility by a prime theorem says that every integer greater...
 4.4.119: The unique factorization of integers theorem says that any integer ...
 4.4.120: Give a reason for your answer. Assume that all variables represent ...
 4.4.121: Give a reason for your answer. Assume that all variables represent ...
 4.4.122: Give a reason for your answer. Assume that all variables represent ...
 4.4.123: Give a reason for your answer. Assume that all variables represent ...
 4.4.124: Give a reason for your answer. Assume that all variables represent ...
 4.4.125: Give a reason for your answer. Assume that all variables represent ...
 4.4.126: Give a reason for your answer. Assume that all variables represent ...
 4.4.127: Give a reason for your answer. Assume that all variables represent ...
 4.4.128: Give a reason for your answer. Assume that all variables represent ...
 4.4.129: Give a reason for your answer. Assume that all variables represent ...
 4.4.130: Give a reason for your answer. Assume that all variables represent ...
 4.4.131: Give a reason for your answer. Assume that all variables represent ...
 4.4.132: Give a reason for your answer. Assume that all variables represent ...
 4.4.133: Fill in the blanks in the following proof that for all integers a a...
 4.4.134: Provestatementsdirectlyfromthedenitionofdivisibility. For all integ...
 4.4.135: Provestatementsdirectlyfromthedenitionofdivisibility. Forallinteger...
 4.4.136: Considerthefollowingstatement:Thenegativeofanymultiple of 3 is a mu...
 4.4.137: Show that the following statement is false: For all integers a and ...
 4.4.138: Determine whether the statement is true or false. Prove the stateme...
 4.4.139: Determine whether the statement is true or false. Prove the stateme...
 4.4.140: Determine whether the statement is true or false. Prove the stateme...
 4.4.141: Determine whether the statement is true or false. Prove the stateme...
 4.4.142: Determine whether the statement is true or false. Prove the stateme...
 4.4.143: Determine whether the statement is true or false. Prove the stateme...
 4.4.144: Determine whether the statement is true or false. Prove the stateme...
 4.4.145: Determine whether the statement is true or false. Prove the stateme...
 4.4.146: Determine whether the statement is true or false. Prove the stateme...
 4.4.147: Determine whether the statement is true or false. Prove the stateme...
 4.4.148: Determine whether the statement is true or false. Prove the stateme...
 4.4.149: Determine whether the statement is true or false. Prove the stateme...
 4.4.150: Determine whether the statement is true or false. Prove the stateme...
 4.4.151: A fastfood chain has a contest in which a card with numbers on it ...
 4.4.152: Is it possible to have a combination of nickels, dimes, and quarter...
 4.4.153: Is it possible to have 50 coins, made up of pennies, dimes, and qua...
 4.4.154: Two athletes run a circular track at a steady pace so that the rst ...
 4.4.155: It can be shown (see exercises 4448) that an integer is divisibleby...
 4.4.156: Use the unique factorization theorem to write the following integer...
 4.4.157: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.4.158: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.4.159: a. If a and b are integers and 12a =25b, does 12b? does 25a? Expl...
 4.4.160: How many zeros are at the end of 458885? Explain how you can answer...
 4.4.161: If n is an integer and n > 1, then n!is the product of n and every ...
 4.4.162: In a certain town 2/3 of the adult men are married to 3/5 of the ad...
 4.4.163: Prove that if n is any nonnegative integer whose decimal representa...
 4.4.164: Prove that if n is any nonnegative integer whose decimal representa...
 4.4.165: Prove that if the decimal representation of a nonnegative integer n...
 4.4.166: Observe that 7524=71000+5100+210+4 =7(999+1)+5(99+1)+2(9+1)+4 = (79...
 4.4.167: Prove that for any nonnegative integer n, if the sum of the digits ...
 4.4.168: Givenapositiveintegern writtenindecimalform,thealternatingsumofthed...
 4.4.169: The quotientremainder theorem says that for all integers n and d w...
 4.4.170: If n and d are integers with d > 0, n div d is _____ and n mod d is...
 4.4.171: The parity of an integer indicates whether the integer is _____.
 4.4.172: According to the quotientremainder theorem, if an integer n is div...
 4.4.173: To prove a statement of the form If A1 or A2 or A3, then C, prove _...
 4.4.174: The triangle inequality says that for all real numbers x and y, _____.
 4.4.175: For each of the values of n and d given, nd integers q and r such t...
 4.4.176: For each of the values of n and d given, nd integers q and r such t...
 4.4.177: For each of the values of n and d given, nd integers q and r such t...
 4.4.178: For each of the values of n and d given, nd integers q and r such t...
 4.4.179: For each of the values of n and d given, nd integers q and r such t...
 4.4.180: For each of the values of n and d given, nd integers q and r such t...
 4.4.181: Evaluate the expressions . a. 43 div 9 b. 43 mod 9
 4.4.182: Evaluate the expressions . a. 50 div 7 b. 50 mod 7
 4.4.183: Evaluate the expressions . a. 28 div 5 b. 28 mod 5
 4.4.184: Evaluate the expressions . a. 30 div 2 b. 30 mod 2
 4.4.185: Check the correctness of formula (4.4.1) given in Example 4.4.3 for...
 4.4.186: Justify formula (4.4.1) for general values of DayT and N.
 4.4.187: On a Monday a friend says he will meet you again in 30 days. What d...
 4.4.188: If today is Tuesday, what day of the week will it be 1,000 days fro...
 4.4.189: January 1, 2000, was a Saturday, and 2000 was a leap year. What day...
 4.4.190: Suppose d is a positive integer and n is any integer. If dn, what ...
 4.4.191: Prove that the product of any two consecutive integers is even.
 4.4.192: The result of exercise 17 suggests that the second apparent blind a...
 4.4.193: Prove that for all integers n,n2 n+3 is odd.
 4.4.194: Suppose a is an integer. If a mod7=4, what is 5a mod7? In other wor...
 4.4.195: Suppose b is an integer. If b mod12=5, what is 8b mod12? In other w...
 4.4.196: Suppose c is an integer. If c mod15=3, what is 10c mod15? In other ...
 4.4.197: Prove that for all integers n, ifn mod5=3 then n2 mod 5=4.
 4.4.198: Prove that for all integers m and n, ifm mod5=2 and n mod5=1 thenmn...
 4.4.199: Prove that for all integers a and b, ifa mod7=5 and b mod7=6 thenab...
 4.4.200: Prove that a necessary and sufcient condition for a nonnegative int...
 4.4.201: Show that any integer n can be written in one of the three forms n ...
 4.4.202: a. Usethequotientremaindertheoremwithd =3toprove that the product ...
 4.4.203: a. Usethequotientremaindertheoremwithd =3toprove that the square o...
 4.4.204: a. Usethequotientremaindertheoremwithd =3toprove that the product ...
 4.4.205: You may use the properties. a. Prove that for all integers m and n,...
 4.4.206: You may use the properties. Given any integers a,b, andc, ifab is e...
 4.4.207: You may use the properties. Given any integers a,b, andc, ifab is o...
 4.4.208: Given any integer n, ifn > 3, could n,n + 2, and n + 4 all be prime...
 4.4.209: Prove each of the statements. The fourth power of any integer has t...
 4.4.210: Prove each of the statements.The product of any four consecutive in...
 4.4.211: Prove each of the statements. The square of any integer has the for...
 4.4.212: Prove each of the statements.For any integer n, n2 +5 is not divisi...
 4.4.213: Prove each of the statements.The sum of any four consecutive intege...
 4.4.214: Prove each of the statements.For any integer n,n(n2 1)(n+2) is divi...
 4.4.215: Prove each of the statements.For all integers m,m2 =5k, or m2 =5k+1...
 4.4.216: Prove each of the statements.Every prime number except 2 and 3 has ...
 4.4.217: Prove each of the statements.If n is an odd integer, then n4 mod 16=1.
 4.4.218: Prove each of the statements.For all real numbers x and y,xy=xy.
 4.4.219: Prove each of the statements.Forallrealnumbersr andcwithc 0,ifc r c...
 4.4.220: Prove each of the statements.For all real numbers r and c with c 0,...
 4.4.221: To prove a statement by contradiction, you suppose that _____ and y...
 4.4.222: A proof by contraposition of a statement of the form x D, ifP(x) th...
 4.4.223: To prove a statement of the form x D, ifP(x) then Q(x) by contrapos...
 4.4.224: Fill in the blanks in the following proof by contradiction that the...
 4.4.225: Is1 0an irrational number? Explain.
 4.4.226: Use proof by contradiction to show that for all integers n, 3n+2 is...
 4.4.227: Use proof by contradiction to show that for all integers m,7m+4 is ...
 4.4.228: Carefully formulate the negations of each of the statements. Then p...
 4.4.229: Carefully formulate the negations of each of the statements. Then p...
 4.4.230: Carefully formulate the negations of each of the statements. Then p...
 4.4.231: Fill in the blanks for the following proof that the difference of a...
 4.4.232: a. When asked to prove that the difference of any irrational number...
 4.4.233: Prove each statement by contradiction. The square root of any irrat...
 4.4.234: Prove each statement by contradiction. The product of any nonzero r...
 4.4.235: Prove each statement by contradiction. Ifa andb arerationalnumbers,...
 4.4.236: Prove each statement by contradiction. For any integer n,n2 2 is no...
 4.4.237: Prove each statement by contradiction. For all prime numbers a,b, a...
 4.4.238: Prove each statement by contradiction. If a,b, and c are integers a...
 4.4.239: Prove each statement by contradiction. For all odd integers a,b, an...
 4.4.240: Prove each statement by contradiction. For all integers a, ifa mod6...
 4.4.241: Fill in the blanks in the following proof by contraposition that fo...
 4.4.242: Prove the statements by contraposition. If a product of two positiv...
 4.4.243: Prove the statements by contraposition. If a sum of two real number...
 4.4.244: Consider the statement For all integers n, ifn2 is odd then n is od...
 4.4.245: Consider the statement For all real numbersr, ifr2 is irrational th...
 4.4.246: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.247: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.248: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.249: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.250: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.251: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.252: Prove each of the statements in two ways: (a) by contraposition and...
 4.4.253: Thefollowingproofthateveryintegerisrationalisincorrect. Find the mi...
 4.4.254: a. Prove by contraposition: For all positive integers n,r, and s, i...
 4.4.255: Use the test for primality to determine whether the following numbe...
 4.4.256: The sieve of Eratosthenes, named after its inventor, the Greek scho...
 4.4.257: Use the test for primality and the result of exercise 33 to determi...
 4.4.258: Use proof by contradiction to show that every integer greater than ...
 4.4.259: The ancient Greeks discovered that in a right triangle where both l...
 4.4.260: One way to prove that 2 is an irrational number is to assume that2=...
 4.4.261: One way to prove that there are innitely many prime numbers is to a...
 4.4.262: A calculator display shows that 2=1.414213562, and 1.414213562= 141...
 4.4.263: Example 4.2.1(h) illustrates a technique for showing that any repea...
 4.4.264: Determine which statements are true and which are false. Prove thos...
 4.4.265: Determine which statements are true and which are false. Prove thos...
 4.4.266: Determine which statements are true and which are false. Prove thos...
 4.4.267: Determine which statements are true and which are false. Prove thos...
 4.4.268: Determine which statements are true and which are false. Prove thos...
 4.4.269: Determine which statements are true and which are false. Prove thos...
 4.4.270: Determine which statements are true and which are false. Prove thos...
 4.4.271: Determine which statements are true and which are false. Prove thos...
 4.4.272: Determine which statements are true and which are false. Prove thos...
 4.4.273: Determine which statements are true and which are false. Prove thos...
 4.4.274: Determine which statements are true and which are false. Prove thos...
 4.4.275: Consider the following sentence: If x is rational thenx is irration...
 4.4.276: a. Prove that for all integers a, ifa3 is even then a is even. b. P...
 4.4.277: a. Use proof by contradiction to show that for any integer n, it is...
 4.4.278: Give an example to show that if d is not prime and n2 is divisible ...
 4.4.279: The quotientremainder theorem says not only that there existquotie...
 4.4.280: Prove that5 is irrational.
 4.4.281: Prove that for any integer a,9 (a2 3).
 4.4.282: An alternative proof of the irrationality of 2 counts the number of...
 4.4.283: Use the proof technique illustrated in exercise 21 to prove that if...
 4.4.284: Prove that2+3 is irrational.
 4.4.285: Prove that log5(2) is irrational. (Hint: Use the unique factorisati...
 4.4.286: Let N =2357+1. What remainder is obtained when N is divided by 2? 3...
 4.4.287: Suppose a is an integer and p is a prime number such that pa and p...
 4.4.288: Let p1, p2, p3,...be a list of all prime numbers in ascending order...
 4.4.289: Use the fact that for all integers n, n!=n(n1)...321. An alternativ...
 4.4.290: Use the fact that for all integers n, n!=n(n1)...321. Prove that fo...
 4.4.291: Prove that if p1, p2,...,and pn are distinct prime numbers with p1 ...
 4.4.292: a. Fermats last theorem says that for all integers n > 2, the equat...
 4.4.293: Note that to show there is a unique object with a certain property,...
 4.4.294: Note that to show there is a unique object with a certain property,...
 4.4.295: Note that to show there is a unique object with a certain property,...
 4.4.296: Note that to show there is a unique object with a certain property,...
Solutions for Chapter 4: Elementary Number Theory and Methods of Proof
Full solutions for Discrete Mathematics: Introduction to Mathematical Reasoning  1st Edition
ISBN: 9780495826170
Solutions for Chapter 4: Elementary Number Theory and Methods of Proof
Get Full SolutionsSince 296 problems in chapter 4: Elementary Number Theory and Methods of Proof have been answered, more than 17120 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics: Introduction to Mathematical Reasoning, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics: Introduction to Mathematical Reasoning was written by and is associated to the ISBN: 9780495826170. Chapter 4: Elementary Number Theory and Methods of Proof includes 296 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).