Precalculus With Limits A Graphing Approach 5th Edition - Solutions by Chapter

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!

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.

peA) = det(A - AI) has peA) = zero matrix.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

space of all combinations of the columns of A.

If diagonalizable, they share n eigenvectors.

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).

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.

A combination other than all Ci = 0 gives L Ci Vi = O.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Appears in block elimination on [~ g ].

Every B = M-I AM has the same eigenvalues as A.

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Orthonormal columns (complex analog of Q).