- 7.1.1: Determine whether or not each of the following matrices is in echel...
- 7.1.2: Determine whether or not each of the following matrices is in echel...
- 7.1.3: In Exercises 3-6, solve the systems of equations using Gaussian eli...
- 7.1.4: In Exercises 3-6, solve the systems of equations using Gaussian eli...
- 7.1.5: In Exercises 3-6, solve the systems of equations using Gaussian eli...
- 7.1.6: In Exercises 3-6, solve the systems of equations using Gaussian eli...
- 7.1.7: Consider a system of four equations in five variables. In general, ...
- 7.1.8: Compare Gaussian elimination to Gauss-Jordan elimination for a syst...
- 7.1.9: Can a matrix have more than one echelon form? (Hint: Consider a 2 X...
- 7.1.10: Consider a system of n linear equations inn variables that has a un...
- 7.1.11: Construct a table that gives the number of multiplications and addi...
Solutions for Chapter 7.1: Gaussian Elimination
Full solutions for Linear Algebra with Applications | 8th Edition
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vector space V.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).