 1.4.1: Label the following statements as true or false. (a) The zero vecto...
 1.4.2: Solve the following systems of linear equations by the method intro...
 1.4.3: For each of the following lists of vectors in R3 , determine whethe...
 1.4.4: For each list of polynomials in P3(K), determine whether the first ...
 1.4.5: In each part, determine whether the given vector is in the span of ...
 1.4.6: Show that the vectors (1,1,0), (1,0,1), and (0,1,1) generate F3 .
 1.4.7: In Fn , let ej denote the vector whose jt h coordinate is 1 and who...
 1.4.8: Show that Pn(F) is generated by {1, x,... , x n }
 1.4.9: Show that the matrices 1 0 0 0 generate M2X2(F)
 1.4.10: Show that if 1 0 0 1 0 0 0 0 1 0 and 0 0 0 1 Mi = 0 0 M2 = 0 0 0 1 ...
 1.4.11: Prove that span({x}) = {ax: a F} for any vector x in a vector space...
 1.4.12: Show that a subset W of a vector space V is a subspace of V if and ...
 1.4.13: Show that if Si and S2 are subsets of a vector space V such that Si...
 1.4.14: Show that if Si and S2 are arbitrary subsets of a vector space V, t...
 1.4.15: Let S\ and S2 be subsets of a vector space V. Prove that span (Si D...
 1.4.16: Let V be a vector space and S a subset of V with the property that ...
 1.4.17: Let W be a subspace of a vector space V. Under what conditions are ...
Solutions for Chapter 1.4: Linear Combinations and Systems of Linear Equations
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 1.4: Linear Combinations and Systems of Linear Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. Chapter 1.4: Linear Combinations and Systems of Linear Equations includes 17 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Since 17 problems in chapter 1.4: Linear Combinations and Systems of Linear Equations have been answered, more than 10864 students have viewed full stepbystep solutions from this chapter.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.