- 10.3.1: The following lines from Book 12 of Homer's Odyssey relate a precur...
- 10.3.2: Solve the following problems from the Bakhshali Manuscript. (a) B p...
- 10.3.3: A problem on a Babylonian tablet requires finding the length and wi...
- 10.3.4: The following two problems are from The Nine Chapters of the Mathem...
- 10.3.5: This problem in part (a) is known as the Flower of Thymaridas, name...
- 10.3.6: For Example 5 from the Bakhshali Manuscript: (a) Express Equations ...
- 10.3.7: Solve the problems posed in the following three epigrams, which app...
- 10.3.T1: (a) Solve Archimedes' Cattle a symbolic algebra program. (b) The Ca...
- 10.3.T2: The following problem is from The Nine Chapters of the Mathematical...
Solutions for Chapter 10.3: The Earliest Applications of Linear Algebra
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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).
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).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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 ().
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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·