- 1.1.1: In each part, determine whether the equation is linear in , , and ....
- 1.1.2: In each part, determine whether the equations form a linear system.
- 1.1.3: In each part, determine whether the equations form a linear system.
- 1.1.4: For each system in Exercise 2 that is linear, determine whether it ...
- 1.1.5: For each system in Exercise 3 that is linear, determine whether it ...
- 1.1.6: Write a system of linear equations consisting of three equations in...
- 1.1.7: In each part, determine whether the given vector is a solution of t...
- 1.1.8: In each part, determine whether the given vector is a solution of t...
- 1.1.9: . In each part, find the solution set of the linear equation by usi...
- 1.1.10: In each part, find the solution set of the linear equation by using...
- 1.1.11: In each part, find a system of linear equations corresponding to th...
- 1.1.12: In each part, find a system of linear equations corresponding to th...
- 1.1.13: In each part, find the augmented matrix for the given system of lin...
- 1.1.14: In each part, find the augmented matrix for the given system of lin...
- 1.1.15: The curve shown in the accompanying figure passes through the point...
- 1.1.16: Explain why each of the three elementary row operations does not af...
- 1.1.17: Show that if the linear equations have the same solution set, then ...
Solutions for Chapter 1.1: Introduction to Systems of Linear Equations
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Change of basis matrix M.
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.)
Column space C (A) =
space of all combinations of the columns of A.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Full row rank r = m.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Skew-symmetric matrix K.
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)!
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.