Linear Algebra

estriction, A-9
right inverse, 231
structure preserving, 161, 165
seehomomorphism, 176
two-sided inverse, 231
well-defined, A-8
zero, 177
Fundamental Theorem
of Linear Algebra, 268
Gauss’ method, 2
accuracy, 67–71
back-substitution, 5
elementary operations, 4
Gauss-Jordan, 46
Gauss-Jordan, 46
generalized nullspace, 367
generalized rangespace, 367
Geometry of Linear Maps, 274–279
Gram-Schmidt process, 255–260
homogeneous coordinate vector, 340
homogeneous coordinates, 291
homogeneous equation, 21
homomorphism, 176
composition, 215
matrix representing, 194–204
nonsingular, 190, 207
nullity, 188
nullspace, 188
rangespace, 184
rank, 206
zero, 177
ideal line, 342
ideal point, 342
identity
function, A-8
matrix, 224
ill-conditioned, 68
implication, A-2
improper subspace, 92
incidence matrix, 228
index
of nilpotency, 370
induced map, 275
induction, 23, A-5
inner product, 39
Input-Output Analysis, 63–66
internal direct sum, 135
invariant
subspace, 377
invariant subspace
definition, 389
inverse, 231, A-9
additive, 80
exists, 231
left, 231, A-9
matrix, 329
right, 231, A-9
two-sided, A-9
inverse function, 231
inverse image, 185
inversion, 313
isometry, 286
isomorphism, 159–175
characterized by dimension, 172
definition, 161
of a space with itself, 163
Jordan block, 388
Jordan form, 379–398
represents similarity classes, 392
kernel, 188
Kirchhoff’s Laws, 73
Klein, F., 286
Laplace expansion, 326–330
computes determinant, 327
leading variable, 5
least squares, 269–273
length, 39
Leontief, W., 63
line
best fit, 269
in projective plane, 341
line at infinity, 342
line of best fit, 269–273
linear
transpose operation, 131
linear combination, 52
Linear Combination Lemma, 52
linear equation, 2
coefficients, 2
constant, 2
homogeneous, 21
inconsistent systems, 269
satisfied by a vector, 15
solution of, 2