 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
ISBN: 9780470432051
Solutions for Chapter 1.1: Introduction to Systems of Linear Equations
Get Full SolutionsChapter 1.1: Introduction to Systems of Linear Equations includes 17 full stepbystep solutions. Since 17 problems in chapter 1.1: Introduction to Systems of Linear Equations have been answered, more than 13824 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
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.

Elimination.
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 edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. 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.

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

Spanning set.
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.