Solutions for Chapter 3.3: Discrete Mathematics and Its Applications 7th Edition

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

space of all combinations of the columns of A.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Vector addition and scalar multiplication.

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

The columns of N are the n - r special solutions to As = O.

Every v in V is orthogonal to every w in W.

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

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

(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.

The right side b is in the column space of A.

Real symmetric A has real A'S and orthonormal q's.

Any vector space inside V, including V and Z = {zero vector only}.