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

peA) = det(A - AI) has peA) = zero matrix.

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Triangular matrix with one extra nonzero adjacent diagonal.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

Square root of x T x (Pythagoras in n dimensions).

A combination other than all Ci = 0 gives L Ci Vi = O.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Constant down each diagonal = time-invariant (shift-invariant) filter.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.