 4.5.1: Show that
 4.5.2: Show that is a linearly independent sel in R3.
 4.5.3: Determine whether IS a linearly independem set in R4
 4.5.4: Determine whether s = l[l ,].[3 8  5].[3 6  9]) IS a linearly in...
 4.5.5: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.6: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.7: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.8: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.9: L" " = [il" = [=} , = m ,"g
 4.5.10: L" " = [H' , = [t}" = m relo"g
 4.5.11: Which o f the given vectors in RJ are linearly JepenJent? For those...
 4.5.12: Consider the vector space ~:. Follow the directions of Exercise II....
 4.5.13: Consider the vector space 1'2. Follow the directims o f Exercise II...
 4.5.14: Let V be the vector space of all realvalued continuous function s....
 4.5.15: Consider the vector nce RJ. Fo llow the directions of Exercise II. ...
 4.5.16: For what values of c are the veCIOl'S [I 0  I J. 12 2], and [ I c...
 4.5.17: For wh:1I vn lu e~ of (' [.re the veCIOl'S t + 3 and 21 + c1 + 2 in...
 4.5.18: Lei u and v be nonzero veCIOrs in a vector ~pace V. Show thai u and...
 4.5.19: Let S = {VI. vl .' ,. \'.1 1 be a SCI of vectors in a vector space ...
 4.5.20: Suppose that S = l VI. v!, vJ! is a line3r1y ind
 4.5.21: Suppose that S = (VI. '0' 2. '0'3) is a linearly independent set of...
 4.5.22: uppose that S = {VI. '0'2. \ ') 1 is a linearly dependent set of ve...
 4.5.23: Show that if (VI. V2) is lineally independelll and v] does aot belo...
 4.5.24: Suppose that {VI. '0'2 ..... "N} is a linearly independent set of v...
 4.5.25: Let A be an //I x /I matrix in reduced row echelon form. Prove that...
 4.5.26: Let S = {UI. U2 ..... UI'} be a set of vectors in a vector space an...
 4.5.27: Let Sj and S2 be finite subsets of a vector space and let 51 be a s...
 4.5.28: Let SI and S2 be finite subsets of a vector space and let 51 be a s...
 4.5.29: Let A be an 1/1 x /I matrix. Associate with A the vector win RN '" ...
 4.5.30: As noted in the Remark after Example 7 in Section 4.4. to detennine...
 4.5.31: (\Vurning: TIle stratcgy given in Exercisc 30 assumes the computati...
Solutions for Chapter 4.5: Linear Independence
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 4.5: Linear Independence
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 4.5: Linear Independence includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 4.5: Linear Independence have been answered, more than 8450 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).