Elementary Linear Algebra with Applications 9th Edition - Solutions by Chapter

A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

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

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

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

= 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.

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

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

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

The rows (or the columns) of A generate a box with volume I det(A) I.