 1.1.2.1: In Exercises 16, find u # v. 1. u c d 12d , v c 31d
 1.1.3.1: write the equation of the line passingthrough P with normal vector ...
 1.1.4.1: determine the resultant of the given forces. f1 acting due north wi...
 1.1.1.1: Draw the following vectors in standard positionin 2:
 1.1.2.2: In Exercises 16, find u # v. 2.u c 23d , v c 96u c d
 1.1.3.2: write the equation of the line passingthrough P with normal vector ...
 1.1.4.2: determine the resultant of the given forces. f1 acting due west wit...
 1.1.1.2: Draw the vectors in Exercise 1 with their tails at thepoint (1, 3).
 1.1.2.3: In Exercises 16, find u # v. 3.u 123 , v 231
 1.1.3.3: write the equation of the line passingthrough P with direction vect...
 1.1.4.3: determine the resultant of the given forces. f1 acting with a magni...
 1.1.1.3: Draw the following vectors in standard positionin 3:(a) a [0, 2, 0]...
 1.1.2.4: In Exercises 16, find u # v. 4.u 1.50.42.1 , v 3.05.20.6
 1.1.3.4: write the equation of the line passingthrough P with direction vect...
 1.1.4.4: determine the resultant of the given forces. 4. f1 acting with a ma...
 1.1.1.4: If the vectors in Exercise 3 are translated so that theirheads are ...
 1.1.2.5: In Exercises 16, find u # v. 5.u 31, 12, 13, 04, v 34, 12, 0, 54
 1.1.3.5: write the equation of the line passingthrough P with direction vect...
 1.1.4.5: determine the resultant of the given forces.5. f1 acting due east w...
 1.1.1.5: For each of the following pairs of points, draw thevector Then comp...
 1.1.2.6: In Exercises 16, find u # v. 6. v 32.29, 1.72, 4.33, 1.544u 31.12, ...
 1.1.3.6: write the equation of the line passingthrough P with direction vect...
 1.1.4.6: determine the resultant of the given forces.f1 acting due east with...
 1.1.1.6: 6. A hiker walks 4 km north and then 5 km northeast.Draw displaceme...
 1.1.2.7: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.7: write the equation of the plane passingthrough P with normal vector...
 1.1.4.7: Resolve a force of 10 N into two forces perpendicularto each other ...
 1.1.1.7: Exercises 710 refer to the vectors in Exercise 1. Computethe indica...
 1.1.2.8: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.8: write the equation of the plane passingthrough P with normal vector...
 1.1.4.8: A 10 kg block lies on a ramp that is inclined at an angleof 30o (Fi...
 1.1.1.8: Exercises 710 refer to the vectors in Exercise 1. Computethe indica...
 1.1.2.9: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.9: write the equation of the plane passingthrough P with direction vec...
 1.1.4.9: A tow truck is towing a car. The tension in the towcable is 1500 N ...
 1.1.1.9: Exercises 710 refer to the vectors in Exercise 1. Computethe indica...
 1.1.2.10: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.10: write the equation of the plane passingthrough P with direction vec...
 1.1.4.10: A lawn mower has a mass of 30 kg. It is being pushedwith a force of...
 1.1.1.10: Exercises 710 refer to the vectors in Exercise 1. Computethe indica...
 1.1.2.11: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.11: give the vector equation of the linepassing through P and Q.11. P (...
 1.1.4.11: A sign hanging outside Joes Diner has a mass of 50 kg(Figure 1.83)....
 1.1.1.11: Exercises 11 and 12 refer to the vectors in Exercise 3.Compute the ...
 1.1.2.12: In Exercises 712, find for the given exercise, and give aunit vecto...
 1.1.3.12: give the vector equation of the linepassing through P and Q.12. P (...
 1.1.4.12: A sign hanging in the window of Joes Diner has amass of 1 kg. If th...
 1.1.1.12: Exercises 11 and 12 refer to the vectors in Exercise 3.Compute the ...
 1.1.2.13: In Exercises 1316, find the distance d(u, v) between u andv in the ...
 1.1.3.13: Give the vector equation of the planepassing through P, Q, and R. 1...
 1.1.4.13: A painting with a mass of 15 kg is suspended by twowires from hooks...
 1.1.1.13: Find the components of the vectors u, v, u v, andu v, where u and v...
 1.1.2.14: In Exercises 1316, find the distance d(u, v) between u andv in the ...
 1.1.3.14: Give the vector equation of the planepassing through P, Q, and R. 1...
 1.1.4.14: A painting with a mass of 20 kg is suspended by twowires from a cei...
 1.1.1.14: In Figure 1.24, A, B, C, D, E, and F are the vertices of aregular h...
 1.1.2.15: In Exercises 1316, find the distance d(u, v) between u andv in the ...
 1.1.3.15: Find parametric equations and an equation in vectorform for the lin...
 1.1.4.15: find the parity check code vector forthe binary vector u. u [1, 0, ...
 1.1.1.15: In Exercises 15 and 16, simplify the given vector expression.Indica...
 1.1.2.16: In Exercises 1316, find the distance d(u, v) between u andv in the ...
 1.1.3.16: Consider the vector equation x p t(q p), wherep and q correspond to...
 1.1.4.16: find the parity check code vector forthe binary vector u. u [1, 1, ...
 1.1.1.16: In Exercises 15 and 16, simplify the given vector expression.Indica...
 1.1.2.17: If u, v, and w are vectors in n, n 2, and c is ascalar, explain why...
 1.1.3.17: Suggest a vector proof of the fact that, in 2, twolines with slopes...
 1.1.4.17: a parity check code vector v is given.Determine whether a single er...
 1.1.1.17: In Exercises 17 and 18, solve for the vector x in terms of thevecto...
 1.1.2.18: u c 21d , v c 13d
 1.1.3.18: The line passes through the point P (1,1, 1) andhas direction vecto...
 1.1.4.18: a parity check code vector v is given.Determine whether a single er...
 1.1.1.18: In Exercises 17 and 18, solve for the vector x in terms of thevecto...
 1.1.2.19: u 211 , v 121
 1.1.3.19: The plane 1 has the equation 4x y 5z 2. Foreach of the planes in Ex...
 1.1.4.19: a parity check code vector v is given.Determine whether a single er...
 1.1.1.19: In Exercises 19 and 20, draw the coordinate axes relative to uand v...
 1.1.2.20: u [5, 4,3], v [1,2,1]
 1.1.3.20: Find the vector form of the equation of the line in 2that passes th...
 1.1.4.20: a parity check code vector v is given.Determine whether a single er...
 1.1.1.20: In Exercises 19 and 20, draw the coordinate axes relative to uand v...
 1.1.2.21: u [0.9, 2.1, 1.2], v [4.5, 2.6,0.8]
 1.1.3.21: Find the vector form of the equation of the line in 2that passes th...
 1.1.4.21: refer to check digit codes in which the checkvector is the vector c...
 1.1.1.21: In Exercises 21 and 22, draw the standard coordinate axes onthe sam...
 1.1.2.22: u [1, 2, 3, 4], v [3, 1, 2,2]
 1.1.3.22: Find the vector form of the equation of the line in 3that passes th...
 1.1.4.22: refer to check digit codes in which the checkvector is the vector c...
 1.1.1.22: In Exercises 21 and 22, draw the standard coordinate axes onthe sam...
 1.1.2.23: u [1, 2, 3, 4], v [5, 6, 7, 8]
 1.1.3.23: Find the vector form of the equation of the line in 3that passes th...
 1.1.4.23: refer to check digit codes in which the checkvector is the vector c...
 1.1.1.23: Draw diagrams to illustrate properties (d) and (e) ofTheorem 1.1.
 1.1.2.24: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.24: Find the normal form of the equation of the plane thatpasses throug...
 1.1.4.24: refer to check digit codes in which the checkvector is the vector c...
 1.1.1.24: Give algebraic proofs of properties (d) through (g) ofTheorem 1.1
 1.1.2.25: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.25: A cube has vertices at the eight points (x, y, z), whereeach of x, ...
 1.1.4.25: Prove that for any positive integers m and n, thecheck digit code i...
 1.1.1.25: In Exercises 2528, u and v are binary vectors. Find u vand in each ...
 1.1.2.26: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.26: Find the equation of the set of all points that areequidistant from...
 1.1.4.26: find the check digit d in the givenUniversal Product Code.26. [0, 5...
 1.1.1.26: In Exercises 2528, u and v are binary vectors. Find u vand in each ...
 1.1.2.27: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.27: Q (2, 2), with equation c xyd c12d t c 11d
 1.1.4.27: find the check digit d in the givenUniversal Product Code.27. [0, 1...
 1.1.1.27: In Exercises 2528, u and v are binary vectors. Find u vand in each ...
 1.1.2.28: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.28: Q (0, 1, 0), with equation xyz 111 t 203
 1.1.4.28: Consider the UPC [0, 4, 6, 9, 5, 6, 1, 8, 2, 0, 1, 5].(a) Show that...
 1.1.1.28: In Exercises 2528, u and v are binary vectors. Find u vand in each ...
 1.1.2.29: In Exercises 2429, find the angle between u and v in thegiven exerc...
 1.1.3.29: find the distance from the point Q tothe plane 29. Q (2, 2, 2), wit...
 1.1.4.29: Prove that the Universal Product Code will detect allsingle errors.
 1.1.1.29: Write out the addition and multiplication tables for 4.
 1.1.2.30: Let A (3, 2), B (1, 0), and C (4, 6). Prove thatABC is a rightangl...
 1.1.3.30: find the distance from the point Q tothe plane 30. Q (0, 0, 0), wit...
 1.1.4.30: (a) Prove that if a transposition error is made in thesecond and th...
 1.1.1.30: Write out the addition and multiplication tables for 5.
 1.1.2.31: Let A(1, 1,1), B(3, 2,2), and C(2, 2,4).Prove that ABC is a righta...
 1.1.3.31: Find the point R on that is closest to Q in Exercise 27.
 1.1.4.31: find the check digit d in the givenInternational Standard Book Numb...
 1.1.1.31: In Exercises 3143, perform the indicated calculations. 31. 2 2 2 in 3
 1.1.2.32: Find the angle between a diagonal of a cube and anadjacent edge.
 1.1.3.32: Find the point R on that is closest to Q in Exercise 28.
 1.1.4.32: Consider the ISBN10 [0, 4, 4, 9, 5, 0, 8, 3, 5, 6].(a) Show that t...
 1.1.1.32: In Exercises 3143, perform the indicated calculations.32.2 # 2 # 2 ...
 1.1.2.33: A cube has four diagonals. Show that no two of themare perpendicular.
 1.1.3.33: Find the point R on that is closest to Q in Exercise 29
 1.1.4.33: Consider the ISBN10 [0, 4, 4, 9, 5, 0, 8, 3, 5, 6].(a) Show that t...
 1.1.1.33: In Exercises 3143, perform the indicated calculations.33.212 1 22 in 3
 1.1.2.34: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.34: Find the point R on that is closest to Q in Exercise 30.
 1.1.4.34: a) Prove that if a transposition error is made in thefourth and fif...
 1.1.1.34: In Exercises 3143, perform the indicated calculations.34.3 1 2 3 in 4
 1.1.2.35: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.35: find the distance between the parallellines. 35. c xyd c 11d s c23d...
 1.1.4.35: Consider the ISBN10 [0, 8, 3, 7, 0, 9, 9, 0, 2, 6].(a) Show that t...
 1.1.1.35: In Exercises 3143, perform the indicated calculations.35.2 # 3 # 2 ...
 1.1.1.36: In Exercises 3143, perform the indicated calculations.36.313 3 22 in 4
 1.1.2.36: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.36: xyz 101 s 111 and xyz 011 t 111
 1.1.1.37: In Exercises 3143, perform the indicated calculations.37.2 1 2 2 1 ...
 1.1.2.37: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.37: find the distance between the parallelplanes. 37. 2x y 2z 0 and 2x ...
 1.1.1.38: In Exercises 3143, perform the indicated calculations.38.13 42 13 2...
 1.1.2.38: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.38: find the distance between the parallelplanes. 38. x y z 1 and x y z 3
 1.1.1.39: In Exercises 3143, perform the indicated calculations.39.816 4 32 in 9
 1.1.2.39: In Exercises 3439, find the projection of v onto u. Draw asketch in...
 1.1.3.39: Prove equation (3) on page 43.
 1.1.1.40: In Exercises 3143, perform the indicated calculations.40.2100 in 11
 1.1.2.40: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.40: Prove equation (4) on page 44.
 1.1.1.41: In Exercises 3143, perform the indicated calculations.41.2, 1, 24 3...
 1.1.2.41: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.41: Prove that, in 2, the distance between parallel lineswith equations...
 1.1.1.42: In Exercises 3143, perform the indicated calculations.42.32, 1, 24 ...
 1.1.2.42: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.42: Prove that the distance between parallel planes withequations d1 an...
 1.1.1.43: In Exercises 3143, perform the indicated calculations.4332, 0, 3, 2...
 1.1.2.43: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.43: find the acute angle between the planeswith the given equations. 43...
 1.1.1.44: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.44: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.44: find the acute angle between the planeswith the given equations.44....
 1.1.1.45: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.45: In Exercises 4045, find the projection of v onto u. Draw asketch in...
 1.1.3.45: show that the plane and line with thegiven equations intersect, and...
 1.1.1.46: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.46: In Exercises 46 and 47, compute the area of the triangle withthe gi...
 1.1.3.46: show that the plane and line with thegiven equations intersect, and...
 1.1.1.47: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.47: In Exercises 46 and 47, compute the area of the triangle withthe gi...
 1.1.3.47: explore one approach to the problem of findingthe projection of a v...
 1.1.1.48: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.48: In Exercises 48 and 49, find all values of the scalar k forwhich th...
 1.1.3.48: explore one approach to the problem of findingthe projection of a v...
 1.1.1.49: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.49: In Exercises 48 and 49, find all values of the scalar k forwhich th...
 1.1.1.50: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.50: Describe all vectors v c xyd v c xyd v c xydu c 31d
 1.1.1.51: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.51: Describe all vectors that are orthogonalto .
 1.1.1.52: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.52: Under what conditions are the following true forvectors u and v in ...
 1.1.1.53: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.53: Prove Theorem 1.2(b).
 1.1.1.54: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.54: Prove Theorem 1.2(d).
 1.1.1.55: In Exercises 4455, solve the given equation or indicate thatthere i...
 1.1.2.55: prove the stated property of distancebetween vectors. 55. d(u, v) d...
 1.1.1.56: 56.(a) For which values of a does x a 0 have a solutionin 5?(b) For...
 1.1.2.56: prove the stated property of distancebetween vectors. 56. d(u, w) d...
 1.1.1.57: 57. (a) For which values of a does ax1 have a solution in5?(b) For ...
 1.1.2.57: prove the stated property of distancebetween vectors. 57. d(u, v) 0...
 1.1.2.58: Prove that u c v c(u v) for all vectors u and v in nand all scalars c.
 1.1.2.59: Prove that u v u v for all vectors u andv in n. [Hint: Replace u by...
 1.1.2.60: Suppose we know that u vu w. Does it followthat v w? If it does, gi...
 1.1.2.61: Prove that (u v) (u v) u2 v2 for all vectorsu and v in n.
 1.1.2.62: (a) Prove that u v2 u v2 2u2 2v2for all vectors u and v in n.(b) Dr...
 1.1.2.63: Prove that for allvectors u and v in n.14u # v 7 u v7 2147 u v7
 1.1.2.64: (a) Prove that u v u v if and only if u and vare orthogonal.(b) Dra...
 1.1.2.65: (a) Prove that u v and u v are orthogonal in n ifand only if u v.(b...
 1.1.2.66: If u 2, v andu v 1, find u v.
 1.1.2.67: Show that there are no vectors u and v such that u1,v2, andu v3.
 1.1.2.68: (a) Prove that if u is orthogonal to both v and w, thenu is orthogo...
 1.1.2.69: Prove that u is orthogonal to v proju(v) for allvectors u and v in ...
 1.1.2.70: (a) Prove that proju(proju(v)) proju(v).(b) Prove that proju(v proj...
 1.1.2.71: The CauchySchwarz Inequality u v u v isequivalent to the inequalit...
 1.1.2.72: Another approach to the proof of the CauchySchwarzInequality is su...
 1.1.2.73: Use the fact that proju(v) cu for some scalar c, togetherwith Figur...
 1.1.2.74: Using mathematical induction, prove the followinggeneralization of ...
Solutions for Chapter 1: Vectors
Full solutions for Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)  3rd Edition
ISBN: 9780538735452
Solutions for Chapter 1: Vectors
Get Full SolutionsChapter 1: Vectors includes 214 full stepbystep solutions. Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) was written by and is associated to the ISBN: 9780538735452. Since 214 problems in chapter 1: Vectors have been answered, more than 12422 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign), edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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].

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.