Explorations in College Algebra 5th Edition - Solutions by Chapter

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

(Particular x p) + (x n in nullspace).

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

dim(V) = number of vectors in any basis for V.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

Complex analog a j i = aU of a symmetric matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

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.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Any solution to Ax = b; often x p has free variables = o.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

A square matrix that has no inverse: det(A) = o.

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

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.