College Algebra 7th Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

Upper triangular systems are solved in reverse order Xn to Xl.

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!

If diagonalizable, they share n eigenvectors.

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

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

Use AT for complex A.

Columns without pivots; these are combinations of earlier columns.

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.

Constant along each antidiagonal; hij depends on i + j.

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

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

Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

For matrix norms II A + B II < II A II + II B II·