 4.5.1: For 110, determine whether the given set of vectors is linearly ind...
 4.5.2: For 110, determine whether the given set of vectors is linearly ind...
 4.5.3: For 110, determine whether the given set of vectors is linearly ind...
 4.5.4: For 110, determine whether the given set of vectors is linearly ind...
 4.5.5: For 110, determine whether the given set of vectors is linearly ind...
 4.5.6: For 110, determine whether the given set of vectors is linearly ind...
 4.5.7: For 110, determine whether the given set of vectors is linearly ind...
 4.5.8: For 110, determine whether the given set of vectors is linearly ind...
 4.5.9: For 110, determine whether the given set of vectors is linearly ind...
 4.5.10: For 110, determine whether the given set of vectors is linearly ind...
 4.5.11: Let v1 = (1, 2, 3), v2 = (4, 5, 6), v3 = (7, 8, 9). Determine wheth...
 4.5.12: Consider the vectors v1 = (2, 1, 5), v2 = (1, 3, 4), v3 = (3, 9, 12...
 4.5.13: Determine all values of the constant k for which the vectors (1, 1,...
 4.5.14: For 1415, determine all values of the constant k for which the give...
 4.5.15: For 1415, determine all values of the constant k for which the give...
 4.5.16: For 1618, determine whether the given set of vectors is linearly in...
 4.5.17: For 1618, determine whether the given set of vectors is linearly in...
 4.5.18: For 1618, determine whether the given set of vectors is linearly in...
 4.5.19: For 1922, determine whether the given set of vectors is linearly in...
 4.5.20: For 1922, determine whether the given set of vectors is linearly in...
 4.5.21: For 1922, determine whether the given set of vectors is linearly in...
 4.5.22: For 1922, determine whether the given set of vectors is linearly in...
 4.5.23: Show that the vectors p1(x) = a + bx and p2(x) = c + dx are linearl...
 4.5.24: If f1(x) = cos 2x, f2(x) = sin2 x, f3(x) = cos2 x, determine whethe...
 4.5.25: For 2531, determine a linearly independent set of vectors that span...
 4.5.26: For 2531, determine a linearly independent set of vectors that span...
 4.5.27: For 2531, determine a linearly independent set of vectors that span...
 4.5.28: For 2531, determine a linearly independent set of vectors that span...
 4.5.29: For 2531, determine a linearly independent set of vectors that span...
 4.5.30: For 2531, determine a linearly independent set of vectors that span...
 4.5.31: For 2531, determine a linearly independent set of vectors that span...
 4.5.32: For 3236, use the Wronskian to show that the given functions are li...
 4.5.33: For 3236, use the Wronskian to show that the given functions are li...
 4.5.34: For 3236, use the Wronskian to show that the given functions are li...
 4.5.35: For 3236, use the Wronskian to show that the given functions are li...
 4.5.36: For 3236, use the Wronskian to show that the given functions are li...
 4.5.37: For 3739, show that the Wronskian of the given functions is identic...
 4.5.38: For 3739, show that the Wronskian of the given functions is identic...
 4.5.39: For 3739, show that the Wronskian of the given functions is identic...
 4.5.40: Consider the functions f1(x) = x, f2(x) = x, if x 0, x, if x < 0. (...
 4.5.41: Determine whether the functions f1(x) = x, f2(x) = x, if x = 0, 1, ...
 4.5.42: Show that the functions f1(x) = x 1, if x 1, 2(x 1), if x < 1, f2(x...
 4.5.43: (a) Show that {1, x, x2, x3} is linearly independent on every inter...
 4.5.44: (a) Show that the functions f1(x) = er1x , f2(x) = er2x , f3(x) = e...
 4.5.45: Let {v1, v2} be a linearly independent set in a vector space V, and...
 4.5.46: If v1 and v2 are vectors in a vector space V, and u1, u2, u3 are ea...
 4.5.47: Let v1, v2,..., vm be a set of linearly independent vectors in a ve...
 4.5.48: Prove from Definition 4.5.4 that if {v1, v2,..., vn} is linearly in...
 4.5.49: Prove that if {v1, v2} is linearly independent and v3 is not in spa...
 4.5.50: Generalizing the previous exercise, prove that if {v1, v2,..., vk }...
 4.5.51: Prove Theorem 4.5.2.
 4.5.52: Prove Proposition 4.5.8.
 4.5.53: Prove that if {v1, v2,..., vk } spans a vector space V, then for ev...
 4.5.54: Prove that if V = Pn(R) and S = {p1, p2,..., pk } is a set of vecto...
Solutions for Chapter 4.5: Linear Dependence and Linear Independence
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 4.5: Linear Dependence and Linear Independence
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. Since 54 problems in chapter 4.5: Linear Dependence and Linear Independence have been answered, more than 21380 students have viewed full stepbystep solutions from this chapter. Chapter 4.5: Linear Dependence and Linear Independence includes 54 full stepbystep solutions.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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