- Chapter 1.1:
- Chapter 1.2:
- Chapter 1.3:
- Chapter 1.4:
- Chapter 1.5:
- Chapter 1.6:
- Chapter 1.7:
- Chapter 1.8:
- Chapter 1.9:
- Chapter 2.1:
- Chapter 2.2:
- Chapter 3.1:
- Chapter 3.2:
- Chapter 3.8:
- Chapter 3.9:
- Chapter 4.1:
- Chapter 4.2:
- Chapter A.1:
- Chapter A.10:
- Chapter A.2:
- Chapter A.3:
- Chapter A.4:
- Chapter A.6:
- Chapter A.7:
- Chapter A.8:
- Chapter A.9:
Introduction to Linear Algebra 5th Edition - Solutions by Chapter
Full solutions for Introduction to Linear Algebra | 5th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).