Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) 3rd Edition - Solutions by Chapter

Tv = Av + Vo = linear transformation plus shift.

If diagonalizable, they share n eigenvectors.

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.

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

Triangular matrix with one extra nonzero adjacent diagonal.

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

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

Vector addition and scalar multiplication.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

= number of pivots = dimension of column space = dimension of row space.

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.

A square matrix that has no inverse: det(A) = o.

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

Spectral radius = max of IAi I.

Columns of n by n identity matrix (written i ,j ,k in R3).

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