Trigonometry 11th Edition - Solutions by Chapter

Remove row i and column j; multiply the determinant by (-I)i + j •

space of all combinations of the columns of A.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

has derivative AeAt; eAt u(O) solves u' = Au.

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Triangular matrix with one extra nonzero adjacent diagonal.

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.

A sequence of steps intended to approach the desired solution.

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.

Nullspace of AT = "left nullspace" of A because y T A = OT.

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

= column times row = rank one matrix.

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.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Orthogonal Q times positive (semi)definite H.

Constant down each diagonal = time-invariant (shift-invariant) filter.

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

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).