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

Parentheses can be removed to leave ABC.

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!

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.

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

Remove row i and column j; multiply the determinant by (-I)i + j •

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Ax = AX with x#-O so det(A - AI) = o.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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

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.

Vector addition and scalar multiplication.

= Xl (column 1) + ... + xn(column n) = combination of columns.

Column and row spaces = lines cu and cv.

Column vectors by convention.

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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

Signs in A = signs in D.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.