- 1.1.1: In each part, determine whether the equation is linear in x1, x2, a...
- 1.1.2: In each part, determine whether the equation is linear in x and y. ...
- 1.1.3: Using the notation of Formula (7), write down a general linear syst...
- 1.1.4: Write down the augmented matrix for each of the linear systems in E...
- 1.1.5: In each part of Exercises 56, find a linear system in the unknowns ...
- 1.1.6: In each part of Exercises 56, find a linear system in the unknowns ...
- 1.1.7: In each part of Exercises 78, find the augmented matrix for the lin...
- 1.1.8: In each part of Exercises 78, find the augmented matrix for the lin...
- 1.1.9: In each part, determine whether the given 3-tuple is a solution of ...
- 1.1.10: In each part, determine whether the given 3-tuple is a solution of ...
- 1.1.11: In each part, solve the linear system, if possible, and use the res...
- 1.1.12: Under what conditions on a and b will the following linear system h...
- 1.1.13: In each part of Exercises 1314, use parametric equations to describ...
- 1.1.14: In each part of Exercises 1314, use parametric equations to describ...
- 1.1.15: In Exercises 1516, each linear system has infinitely many solutions...
- 1.1.16: In Exercises 1516, each linear system has infinitely many solutions...
- 1.1.17: In Exercises 1718, find a single elementary row operation that will...
- 1.1.18: In Exercises 1718, find a single elementary row operation that will...
- 1.1.19: In Exercises 1920, find all values of k for which the given augment...
- 1.1.20: In Exercises 1920, find all values of k for which the given augment...
- 1.1.21: The curve y = ax2 + bx + c shown in the accompanying figure passes ...
- 1.1.22: Explain why each of the three elementary row operations does not af...
- 1.1.23: Show that if the linear equations x1 + kx2 = c and x1 + lx2 = d hav...
- 1.1.24: Consider the system of equations ax + by = k cx + dy = l ex + fy = ...
- 1.1.25: Suppose that a certain diet calls for 7 units of fat, 9 units of pr...
- 1.1.26: Suppose that you want to find values for a, b, and c such that the ...
- 1.1.27: Suppose you are asked to find three real numbers such that the sum ...
Solutions for Chapter 1.1: Introduction to Systems of Linear Equations
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th 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.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
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 .
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).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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).
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vandermonde matrix V.
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.