Algebra 1, Student Edition (MERRILL ALGEBRA 1) 1st Edition - Solutions by Chapter

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

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

The n roots are the eigenvalues of A.

space of all combinations of the columns of A.

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

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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].

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

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Vector addition and scalar multiplication.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

(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 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.

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

Signs in A = signs in D.

Orthonormal columns (complex analog of Q).