Linear Algebra: A Geometric Approach 2nd Edition - Solutions by Chapter

dim(V) = number of vectors in any basis for V.

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Invert A by row operations on [A I] to reach [I A-I].

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.

Complex analog a j i = aU of a symmetric matrix.

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Vector addition and scalar multiplication.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Column and row spaces = lines cu and cv.

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.

Each equation gives a plane in Rn; the planes intersect at x.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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