Index

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adjoint matrix 80

angle between vectors 172

Apollonius' circle 103

Argand diagram 97

asymptotes 143

augmented matrix 2

basis 63

left{to{right 64

Cauchy-Schwarz inequality 165

centroid 192

characteristic equation 122

coe±cient matrix 2, 27

cofactor 78

column vector 27

complex number 91

imaginary part 91

real part 91

real 91

imaginary 92

rationalization 93

complex exponential 110

modulus 101

square root 94

argument 106

complex conjugate 99

modulus-argument form 106

polar representation 106

ratio formulae 103

complex plane 97

consistent system 1, 11

cross-ratio 117

Cramer's rule 40, 86

cosine rule 172

De Moivre's theorem 110

dependent unknowns 11

determinant 74

2 £ 2 matrix 73

lower triangular 76

scalar matrix 76

Laplace expansion 75

diagonal matrix 76

diagonal matrix 50

di®erential equations 125

direction vector 170

distance to a plane 190

dot product 137, 162, 163

eigenvalue 122

eigenvector 122

elementary row matrix 42

elementary row operations 7

ellipse 143, 144

factor theorem 97

¯eld 3

additive inverse 4

multiplicative inverse 4

Gauss' theorem 97

Gauss-Jordan algorithm 8

Gram matrix 138

homogeneous system 16

hyperbola 143

identity matrix 31

imaginary axis 98

inconsistent system 1

independent unknowns 12

194

inversion 76

invertible matrix 37

Joachimsthal 168

least squares

normal equations 48

residuals 48

linear combination 17

linear equation 1

linear transformation 27

re°ection 29

rotation 28

projection 30

linear independence 41, 60

left{to{right test

lower half plane 98

Markov matrix 54

mathematical induction 32

matrix 23

addition 23

additive inverse 24

equality 23

inverse 37

orthogonal 136

power 32

product 25

proper orthogonal 136

rank 68

scalar multiple 24

singular 37

skew{symmetric 47

symmetric 47

subtraction 24

transpose 46

minor 74

modular addition 4

multiplication 4

non{singular matrix 37, 50

non{trivial solution 17

orthogonal

matrix 120, 136

vectors 174

parabola 143

parallel lines 170

vectors 170

parallelogram law 156

perpendicular vectors 174

plane 182

normal form 187

through 3 points 183, 184

position vector 161

positive octant 160

real axis 97

recurrence relations 33, 35

row-echelon form 6

reduced 6

row{equivalence 7

rotation equations 28, 120

scalar triple product 179

skew lines 178

subspace 57

basis 63

column space 58

dimension 65

generated 58

null space 57

row space 58

Surveyor's formula 87

system of equations 1

Three{dimensional space 160

coordinate axes 160

coordinate planes 160

distance 161

line equation 167

projection on a line 177

triangle inequality 166

trivial solution 17

unit vectors 28, 164

upper half plane 98

vector

cross-product 159, 179

195

equality 156, 171

length 137, 157, 164

scalar multiplication 156, 168

of unknowns 27

zero matrix 24

196

SOLUTIONS TO PROBLEMS

ELEMENTARY

LINEAR ALGEBRA

K. R. MATTHEWS

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF QUEENSLAND

First Printing, 1991

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