For all vectors u and v in 'R.". u v = v u. 73. The

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 1- 8, rwo vecror., u and v are given. Compure rile norms of rile vecrors and rire disrance d berween rhem.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Exercises 9- 16, rwo vecrors al'l! given. Compwe lire dol producr of rile vecrors. and detennine wilerher lire vecrors are orrlwgmral.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 17- 24, rwo orrhofimrai vecrors u and v are given. Compure rile quamiries llull 2 llvll 2 , and Jlu + v11 2 . Use your resulrs ro illusrrare rhe Pyrlragorean rileorem.

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 25. u = m and v = [ =~J

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 26. u = m and v = [ _;J

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 27. ~Jandv=[-n

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 28. u = [ -~] and v = m

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 29. u = [ -n and v = [ :]

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 30. u = [ -n and v = [l]

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 31. u = [-n andv = ~]

In Etercises 25- 32. rwo vecrors u and v are given. Compure rire quamiries JluJI, llvll. and Jlu +vii. Use your resulrs ro illusrrare rile rriangie inequaiiry. 32. U = [ -~] and v = [ ~

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 33. u = [ -;J and

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 34. u = m and v =

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 35. u = ] and v

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 36. u = [- ! ]and v =

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 37. u = [-!] andv

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 38. u = [ :] and ' ' =

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 39. u = [n andv=

In Exercises 33-40. two vectors u and v are given. Compllte the qualllities null . Uvll. and u v. Use your results w illustrate the Cauchy Sclnrar~ inequality. 40. u = [ -n and

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 4 1. u = [~] andy = 0

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 42. u = ~] anti y

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 43. u = [!] and y = - x

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 44. u = [!] and y

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 45. u = [~] andy = 3x

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 46. u = [ -;J and y=

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 47. u = m and )' = -3x

In Exercises 41- 48. a ector u and a line Lin n 2 are given Compllle the orthogonal projection w of u on l. and use it to compute the distance d fmm tlw endpni111 of u ltJ C. 48. u = mand y=-4x

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 49. Compute (u + v) . w.

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 50. Compute 114wll .

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 51. Compute llu + vU2

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 52. Compute (u + w) v.

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 53. Compute II'' - 4wll 2.

For Erercises 49- 54, suppose thlll u, v, and w are vectors inn" such that !l ull = 2. nvll = 3. llwll = 5, UV = - I. UW =I. andv w = -4. 54. Compute 112u + 3vU2.

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 55. Compute 11' '11 1

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 56. Compute n3ull .

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 57. Compute v u

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 58. Compute w (u + v).

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 59. Compute 12u .

For Exercises 55- 60. suppose that u. v, and w are vectors inn" .mch 1/101 U oil = 14, U V = 7, U o\1' = - 20, V V = 21, v w = -5, and w . w = 30. 60. Compute [[ v

Vectors must be of the same size for their dot product to be defined.

The dot product of two vectors in n" is a vector in n".

The norm of a vector equals the dot product of the vector with itself.

The norm of a multiple of a vector is the same multiple of the norm of the vector.

The norm of a sum of vectors is the sum of the norms of the vectors.

The squared norm of a sum of 011hogonal vectors is the sum of the squared noms of the vectors.

The orthogonal projection of a vector on a line is a vector that lies along the line.

The norm of a vector is always a nonnegative real number.

If the norm of ,. equals 0. then v = 0.

If u v = 0. then u = 0 or v = 0.

For all vectors u and v in n", [u vi = null nvll

For all vectors u and v in 'R.". u v = v u. 73. The distance between vectors u and ,. in n

The distance between vectors u and ,. in n is Uu - vii .

For all vectors u and v inn" and every scalar c. (cu) v = u (cv).

For all vectors u. v. and w in n". U (\' + w) = liV + UW.

If A is an 11 x 11 matrix and u and v are vectors in n". thenAu v = u. Av.

For every vector v in n", Uvll = II -vii .

If u and v are onhogonal vectors in n". then Uu +vii= Uu ll + Uvll

If w is the orthogonal projection of u on a line through the origin of n2 , then u - w is orthogonal to every vector on the line.

If w is the orthogonal projection of u on a line through the origin of n2 then w is the vector on the line closest to u.

Prove (a) of Theorem 6. 1.

Prove (b) of Theorem 6. 1.

Prove (c) of Theorem 6. 1.

Prove (e) of Theorem 6. 1.

Prove (f) of Theorem 6.1.

Prove that if u is onhogonal to both v and w. then u is orthogonal to every linear combination of v and w.

Let {v, w } be a basis for a subspace W of n", and define \ ' \V z = w - --v. VV Prove that {v. z} is a basis for II' consisting of orthogonal vectors.

Prove that the Cauchy- Schw

Prove that the triangle inequality is an equality if and only if u is a nonnegative multiple of v or v is a nonnegative multiple of u.

Use the lriangle inequalily 10 prove lhal I ll vii- ll wlll:::; ll v - wll for all vectors ' ' and w in R!'.

Prove (u + v) . w = u w + v w for all veclors u, v. and w in R"

Let z be a vector in R". Let W = (u e R": UZ = 0}. Prove that W is a subspace of 7?.".

Let S be a subset of 7?." and W = (u e 7?.": u Z = 0 for all z inS}. Prove that W is a subspace of R".

Let W denote the set of all vectms 1hat lie along 1he line with equalion y = 2x. Find a vee lor z in R 2 such lhat W = (u e 7?.2: u z = 0}. Juslify your answer.

Prove the parallelogram law for veclors in R":

Prove that if u and v are orthogonal nonzero vectors in R". then they are linearly independent.

(a) Prove 1ha1 ATA and A have the same null space. Hi11t: Let v be a vector in 7?." such that AT Av = 0. Observe lhal ATAV V =AvAv = 0. (b) Usc (a) to prove that r.mk AT A = rank A.

Let u and v be nonzero vectors in n 2 or n 3, and lei () be the angle between u and v. Then u, v. and v - u delermine a triangle. (Sec Figure 6.6.) The relationship between the lengths of the sides of this 1riangle and () is called the law of cosi11es. It states lhat Use I he law or cosines and Theorem 6. I 10 derive I he formula UV = ll ullll vJJ cosO.

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 99. u = [-nand v = [i]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 100. ~]andv=[-~]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 101. -~]andv =[-~]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 102. u = [- :] and v = n]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 103. u = ~] and v = [l]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 104. u = [-!] and v = [ ~]

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 105. u = [-~] and v =

/11 Exercises 99-106, use the formula ill Exercise 98to detennine the a11gle betwee11 the vectors u a11d v. 106. u = [ nand V= [l] Ler u a11d v be vectors in 7?.3. Deji11e u x v to be the vecTor ::~:~ = ::~: ]. w

For every vector u in 7?.3. prove lhat u x u = 0.

Prove that u x v = -(v x u) for all vectors u and ' ' in nJ.

For every vector u in 7?.3. prove that u x 0 = 0 x u = 0.

For all vec1ors u and v in 7?.3, prove thai u and v are parallel if and only if u x v = 0.

For all vec1ors u and'' in 7?.3 and all scalars c, prove 1ha1 c(u x v) = c u x v = u x cv.

For all vcclors u. v. and w in 7?.3. prove that ll X (V + w) = II X V + II X \\'.

For all vectors u, v, and win 7?.3, prove that (u + v) X \1' = U X W + V X \1'.

For all veclors u and ,. in 7?.3. prove thai u x v is orthog onal to bolh 11 and v.

For all vectors 11, v, and win 7?.3 , pmve 1hat (u x v) . w = u. (v x w).

For all veclors u. v, and w in 7?.3, prove thai u x (v x w) = (u w)v- (u v)w.

For all vectON u. 1', and w in n3' prove that (u x v) x w = ( w. u)v - (w. v)u.

For all vectON u and v in n J_ prove that Uu x " ' 2 = 1Ju ll2 1vll 2 - (u v) 2

For all vectors u and v in 'R.3 prove that u X ' 'II= t u D' 'll smll. where ll 1s the angle between u and '' Him: u~c Excrci;cs 98 and 118.

For all \'CCtors u. v. and w in R.3 prove the Jacobi idmtity: (U X v ) X W + (\' X w ) X U + ( W X U) X \ " = 0

.1erdw< 121 124 rt'fl'r wtht' IIJ>(IIit"atitm regartli11g the two meth{)(ls of comp11ti11g tl\'eragl' class si:e given i11 this sectio11. Section I cont:lin> 8 student,, section 2 contains 12 stu -dents, and section 3 contains 6 students.

.1erdw< 121 124 rt'fl'r wtht' IIJ>(IIit"atitm regartli11g the two meth{)(ls of comp11ti11g tl\'eragl' class si:e given i11 this sectio11. Section I contains 15 students. and each of sections 2 and 3 cont:lin' JO student.

!11 Eurctses 121 123, tlalll a~ give11 for smdems enrolled i11 a thru-uction semilltll" co11rse. Comp11te the average v determilletl!Jy the .wpe111isor and the average ' ' termi11eti/Jy the illl'e.nigator. Each of the three section> contains 22 students. Section I contains 15 students. and each of sections 2 and 3 cont:lin' JO student.

!11 Eurctses 121 123, tlalll a~ give11 for smdems enrolled i11 a thru-uction semilltll" co11rse. Comp11te the average v determilletl!Jy the .wpe111isor and the average ' ' determi11eti/Jy the illl'e.nigator. Use Exerci>e 88 to prove that the two averaging methods for determining clas> >ize are equal if and only if all of the class ,izes are equal.

In every triangle. the length of any side is less than the sum of the lengths of the other two sides. When this observation is gcncraliu:d ton. we obtain the triallgl~ inequality (Theorem 6.4 ). "hich "Jte' [ u + v ~ l ull+ II,. for any vectors u nnd ,. in n. Let 4.01 ~ [il -[=~] [ 2.01 ] v, = 6.01 . [ 3.0 1] 6.0 1 " 2 = 9.0 1 . 12.0 1 8.01 (a) Verify the triangle inequality for u and '' (b) Verify the tri:mglc inc4uality for u and v1. (c) Verify the triangle inequality for u tmd v2. and (d) From what you have observed in (b) and (c). make a conjecture about when ec1uality occurs in the triangle inequality. (e) lmerpret your conjecture in {tl) geometrically in n 2