 6.6.71: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.79: In each of 1 through 5, write u as a sum of a vector in S and a vec...
 6.6.89: 1 through 4, involve use of the GramSchmidt orthogonalization proce...
 6.6.1: In each of 1 through 5, compute F + G, F G, 2F, 3G, and F .F = 2i 3...
 6.6.16: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.31: In each of 1 through 4, compute FG and GF and verify the anticommut...
 6.6.45: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.72: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.80: In each of 1 through 5, write u as a sum of a vector in S and a vec...
 6.6.90: 1 through 4, involve use of the GramSchmidt orthogonalization proce...
 6.6.2: In each of 1 through 5, compute F + G, F G, 2F, 3G, and F .F = i 3k...
 6.6.17: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.32: In each of 1 through 4, compute FG and GF and verify the anticommut...
 6.6.46: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.73: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.81: In each of 1 through 5, write u as a sum of a vector in S and a vec...
 6.6.91: 1 through 4, involve use of the GramSchmidt orthogonalization proce...
 6.6.3: In each of 1 through 5, compute F + G, F G, 2F, 3G, and F .F = 2i 5...
 6.6.18: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.33: In each of 1 through 4, compute FG and GF and verify the anticommut...
 6.6.47: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.74: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.82: In each of 1 through 5, write u as a sum of a vector in S and a vec...
 6.6.92: 1 through 4, involve use of the GramSchmidt orthogonalization proce...
 6.6.4: In each of 1 through 5, compute F + G, F G, 2F, 3G, and F .F = 2i j...
 6.6.19: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.34: In each of 1 through 4, compute FG and GF and verify the anticommut...
 6.6.48: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.75: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.83: In each of 1 through 5, write u as a sum of a vector in S and a vec...
 6.6.93: Approximate f (x) = x 2 on [0, ] with a linear combination of the f...
 6.6.5: In each of 1 through 5, compute F + G, F G, 2F, 3G, and F
 6.6.20: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.35: In each of 5 through 9, determine whether the points are collinear....
 6.6.49: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.76: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.84: Let S be a subspace of Rn . Determine (S)
 6.6.94: Repeat 5, except now use the functions sin(x), ,sin(5x).
 6.6.6: In each of 6 through 9, find a vector having the given length and i...
 6.6.21: In each of 1 through 6, compute the dot product of the vectors and ...
 6.6.36: In each of 5 through 9, determine whether the points are collinear....
 6.6.50: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.77: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.85: Suppose S is a subspace of Rn . Determine a relationship between th...
 6.6.95: Approximate f (x)= x(2 x) on [2, 2] using a linear combination of t...
 6.6.7: In each of 6 through 9, find a vector having the given length and i...
 6.6.22: In each of 7 through 12, find the equation of the plane containing ...
 6.6.37: In each of 5 through 9, determine whether the points are collinear....
 6.6.51: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.78: In each of 1 through 8, use the GramSchmidt process to find an ort...
 6.6.86: Let S be the subspace of R4 spanned by < 1, 0, 1, 0 > and < 0, 0, 2...
 6.6.8: In each of 6 through 9, find a vector having the given length and i...
 6.6.23: In each of 7 through 12, find the equation of the plane containing ...
 6.6.38: In each of 5 through 9, determine whether the points are collinear....
 6.6.52: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.87: Let S be the subspace of R5 spanned by < 1, 1, 1, 0, 0 >, < 0, 2, 1...
 6.6.9: In each of 6 through 9, find a vector having the given length and i...
 6.6.24: In each of 7 through 12, find the equation of the plane containing ...
 6.6.39: In each of 5 through 9, determine whether the points are collinear....
 6.6.53: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.88: Let S be the subspace of R6 spanned by < 0, 1, 1, 0, 0, 1 >, < 0, 0...
 6.6.10: In each of 10 through 15, find the parametric equations of the line...
 6.6.25: In each of 7 through 12, find the equation of the plane containing ...
 6.6.40: In each of 10, 11, and 12, find a vector normal to the given plane....
 6.6.54: In each of 1 through 10, determine whether the vectors are linearly...
 6.6.11: In each of 10 through 15, find the parametric equations of the line...
 6.6.26: In each of 7 through 12, find the equation of the plane containing ...
 6.6.41: In each of 10, 11, and 12, find a vector normal to the given plane....
 6.6.55: In each of 11 through 15, show that the set S is a subspace of the ...
 6.6.12: In each of 10 through 15, find the parametric equations of the line...
 6.6.27: In each of 7 through 12, find the equation of the plane containing ...
 6.6.42: In each of 10, 11, and 12, find a vector normal to the given plane....
 6.6.56: In each of 11 through 15, show that the set S is a subspace of the ...
 6.6.13: In each of 10 through 15, find the parametric equations of the line...
 6.6.28: In each of 13, 14, and 15, find the projection of v onto u.v = i j ...
 6.6.43: Let F and G be nonparallel vectors and let R be the parallelogram f...
 6.6.57: In each of 11 through 15, show that the set S is a subspace of the ...
 6.6.14: In each of 10 through 15, find the parametric equations of the line...
 6.6.29: In each of 13, 14, and 15, find the projection of v onto u.v = 5i +...
 6.6.44: Form a parallelepiped (skewed rectangular box) having as incident s...
 6.6.58: In each of 11 through 15, show that the set S is a subspace of the ...
 6.6.15: In each of 10 through 15, find the parametric equations of the line...
 6.6.30: In each of 13, 14, and 15, find the projection of v onto u.v = i + ...
 6.6.59: In each of 11 through 15, show that the set S is a subspace of the ...
 6.6.60: In each of 16, 17, and 18, find the coordinates ofX with respect to...
 6.6.61: In each of 16, 17, and 18, find the coordinates ofX with respect to...
 6.6.62: In each of 16, 17, and 18, find the coordinates ofX with respect to...
 6.6.63: Suppose V1, ,Vk form a basis for a subspace S of Rn . Let U be any ...
 6.6.64: Let V1, ,Vk be mutually orthogonal vectors in Rn . Prove that V1 ++...
 6.6.65: Let X and Y be vectors in Rn , and suppose that X= Y . Show that X ...
 6.6.66: Let V1, ,Vk be mutually orthogonal vectors in Rn . Show that, for a...
 6.6.67: Suppose V1, ,Vn are a basis for Rn , consisting of mutually orthogo...
 6.6.68: Show that any finite set of vectors that includes the zero vector i...
 6.6.69: Let S be a nontrivial subspace of Rn . Show that any spanning set o...
 6.6.70: Let u1, , uk be linearly independent vectors in Rn , with k < n. Sh...
Solutions for Chapter 6: Vectors and Vector Spaces
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 6: Vectors and Vector Spaces
Get Full SolutionsChapter 6: Vectors and Vector Spaces includes 95 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since 95 problems in chapter 6: Vectors and Vector Spaces have been answered, more than 7748 students have viewed full stepbystep solutions from this chapter.

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.)

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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 ().

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

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.