- 1.1.1: Use back substitution to solve each of the following systems of equ...
- 1.1.2: Write out the coefficient matrix for each of the systems in Exercis...
- 1.1.3: In each of the following systems, interpret each equation as a line...
- 1.1.4: Write an augmented matrix for each of the systems in Exercise 3.
- 1.1.5: Write out the system of equations that corresponds to each of the f...
- 1.1.6: Solve each of the following systems: (a) x1 2x2 = 5 3x1 + x2 = 1 (b...
- 1.1.7: The two systems 2x1 + x2 = 3 4x1 + 3x2 = 5 and 2x1 + x2 = 1 4x1 + 3...
- 1.1.8: Solve the two systems x1 + 2x2 2x3 = 1 2x1 + 5x2 + x3 = 9 x1 + 3x2 ...
- 1.1.9: Given a system of the form m1x1 + x2 = b1 m2x1 + x2 = b2 where m1, ...
- 1.1.10: Consider a system of the form a11x1 + a12x2 = 0 a21x1 + a22x2 = 0 w...
- 1.1.11: Give a geometrical interpretation of a linear equation in three unk...
Solutions for Chapter 1.1: Systems of Linear Equations
Full solutions for Linear Algebra with Applications | 8th Edition
Basis for V.
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!
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free variable Xi.
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!).
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear transformation T.
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.