 5.1.1: III Hm:isI'I I and 2, find the lengll, of each I'I'C/OI:
 5.1.2: III Hm:isI'I I and 2, find the lengll, of each I'I'C/OI:
 5.1.3: III {ercisl'.Y 3 and 4, compule Ilu  vII.
 5.1.4: III {ercisl'.Y 3 and 4, compule Ilu  vII.
 5.1.5: III lt~rrisl'.\' 5 and 6, find the dis/alice between u and v.
 5.1.6: III lt~rrisl'.\' 5 and 6, find the dis/alice between u and v.
 5.1.7: III t.xerci.\'/'s 7 and 8, deferminl' aI/ \'II/III'S of c j'() Iha...
 5.1.8: III t.xerci.\'/'s 7 and 8, deferminl' aI/ \'II/III'S of c j'() Iha...
 5.1.9: For each pair of vectors u and v in Exercise 5, find the cosine of ...
 5.1.10: For each pair of vectors in Exercise 6, find the cosine of (he angl...
 5.1.11: For each of the following vectors v. find the direction cosines (th...
 5.1.12: Let P and Q be the )Xlints in R3. with respective coordinates (3. ...
 5.1.13: Prove Theorem 5.1.
 5.1.14: Verify Theorem S.I for andc=  3.
 5.1.15: Show that in R2. (a) i  i = j j =l: (b) j.j =O.
 5.1.16: Show that in R3. (a) i  i = j j = k.k = I: (b) i . j = i . k =j ....
 5.1.17: Which of the vectors V I = [~] . v! = [~]. V3 = [ ~]. ~ [;J', ~ [;...
 5.1.18: (a) orthogonal? (b) in the same direction? (e) in op)Xlsite directions
 5.1.19: Whieh of the follm"'ing pairs of lines arc perpendicular? (a) x= 2+...
 5.1.20: Find pammetrie equations of the line passing throuf.!.h (3,  I.  ...
 5.1.21: A ship is being pushed by a tugboat with a force of 300 pounds alon...
 5.1.22: Sup)Xlse that an airplane is flying with an airspeed of 260 kilomet...
 5.1.23: Let points A. B. C. and D in R3 have respective coordinates (1,2,3)...
 5.1.24: Find (. so that the vector V = [~] is orthogonal to
 5.1.25: Find c so tlwt the vector v
 5.1.26: text cannot be copied
 5.1.27: If po,,;h],. hod" ",d b '0 'h" , ~ [~] ;, onhog","] 'obo'h ~ m ,"d ~ m
 5.1.28: Find (' so that the vectors [ ~ ] and [~] are parallel.
 5.1.29: Let (} be the angle between the nonzero vectors u and v in R! or R ...
 5.1.30: Show that the only vector x in R2 or R3 that is orthogonal to every...
 5.1.31: Prove that if v. w. and x are in R2 or R 3 and v is orthogonal to b...
 5.1.32: Let u be a fixed vector in R" (R'). Prove that the set V of all vec...
 5.1.33: Prove that if c is a scalar and v is a vector in R2 or Rl .
 5.1.34: Show that if x is a nonzero vector in R2 or R' . then
 5.1.35: Let S = ( VI. V2. Vl) be a set of nonzero vectors in R' such that a...
 5.1.36: Prove that for any vectors u. \'. and \\. in R2 or Rl . we have u (...
 5.1.37: Prove that for any vectors u. v. and w in R2 or Rl and any scalar c...
 5.1.38: Prove that the diagonals of a rectangle are of equal length. [Him: ...
 5.1.39: Prove that the angles at the base of an isosceles triangle are equal.
 5.1.40: Prove that a parallelogram is a rhombus. a parallelogram with four ...
 5.1.41: To compute the dot product of a pair of vectors u and V in R" or R'...
 5.1.42: Determine whether there is a command in your software to compute th...
 5.1.43: Assuming that your software has a command to conpute the length of ...
 5.1.44: Referring to Exercise 41. how could your software check for orthogo...
Solutions for Chapter 5.1: length and Diredion in R2 and R3
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 5.1: length and Diredion in R2 and R3
Get Full SolutionsChapter 5.1: length and Diredion in R2 and R3 includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 44 problems in chapter 5.1: length and Diredion in R2 and R3 have been answered, more than 9422 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593.

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.

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.

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.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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