Get solution: Exercises 18 refer to the vectors u1 to u8.u1 =312, u2 =111, u3 = 201, u4

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the following dot products. (a) u1 u5 (b) u3 (3u2) (c) u4 u7 (d) 2u4 u7

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the following dot products. (a) 3u7 u3 (b) u1 u1 (c) u2 (2u5) (d) 2u2 (u5)

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the norms of the given vectors. (a) u7 (b) u7 (c) 2u5 (d) 3u5

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the norms of the given vectors. (a) u8 (b) 3u8 (c) u2 (d) 2u2

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the distance between the given vectors. (a) u1 and u2 (b) u3 and u8 (c) 2u6 and u3 (d) 3u2 and 2u5

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Compute the distance between the given vectors. (a) u5 and u1 (b) u2 and u8 (c) 2u3 and u8 (d) 4u2 and 2u6

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Determine if the given vectors are orthogonal. (a) u1 and u3 (b) u3 and u4 (c) u2 and u5 (d) u1 and u8

Exercises 18 refer to the vectors u1 to u8. u1 = 3 1 2  , u2 = 1 1 1  , u3 =  2 0 1  , u4 =  1 3 2  , u5 = 2 1 1  , u6 =  0 3 1  , u7 =  3 4 2  , u8 = 1 1 3  Determine if the given vectors are orthogonal. (a) u2 and u3 (b) u1 and u2 (c) u8 and u5 (d) u3 and u6

For Exercises 912, find all values of a so that u and v are orthogonal. u =  a 2 3  , v

For Exercises 912, find all values of a so that u and v are orthogonal. u = 1 a 5  , v =  7 a 2 

For Exercises 912, find all values of a so that u and v are orthogonal. u = 2 a 3 1 , v = 5 4 6 a

For Exercises 912, find all values of a so that u and v are orthogonal. u = 1 5 a 0 , v = 4 1 a 2

For Exercises 1316, determine if the given vectors form an orthogonal set. u1 =  1 2 , u2 =  4 3

For Exercises 1316, determine if the given vectors form an orthogonal set. u1 = 1 2 3  , u2 =  5 4 1  , u3 =  1 1 1 

For Exercises 1316, determine if the given vectors form an orthogonal set. u1 =  2 2 1  , u2 = 5 13 16 , u3 =  5 4 2 

For Exercises 1316, determine if the given vectors form an orthogonal set. u1 = 1 2 0 1 , u2 = 5 2 4 9 , u3 = 2 2 3 2

In Exercises 1718, find all values of a (if any) so that the given vectors form an orthogonal set. u1 = 1 0 2  , u2 = 4 3 2  , u3 = 6 a 3 

In Exercises 1718, find all values of a (if any) so that the given vectors form an orthogonal set. u1 = 1 3 2 1 , u2 = 4 2 1 0 , u3 = 1 0 a 7

In Exercises 1920, find all values of a and b (if any) so that the given vectors form an orthogonal set. u1 =  2 1 1  , u2 =  3 4 2  , u3 = 2 a b 

In Exercises 1920, find all values of a and b (if any) so that the given vectors form an orthogonal set. u1 = 1 3 6 1 , u2 = 2 1 a 5 , u3 = 0 4 3 b

For Exercises 2124, verify that the Pythagorean Theorem holds for the given orthogonal vectors. u1 =  3 1 , u2 =  1 3

For Exercises 2124, verify that the Pythagorean Theorem holds for the given orthogonal vectors. u1 =  6 8 , u2 =  4 3

For Exercises 2124, verify that the Pythagorean Theorem holds for the given orthogonal vectors. u1 =  2 3 1  , u2 = 4 3 1 

For Exercises 2124, verify that the Pythagorean Theorem holds for the given orthogonal vectors. u1 = 5 2 4 2 , u2 = 2 9 3 2

Suppose that u1 and u2 are orthogonal vectors, with u1 = 2 and u2 = 5. Find 3u1 + 4u2.

Suppose that u1 and u2 are orthogonal vectors, with u1 = 3 and u2 = 4. Find 2u1 u2.

For Exercises 2728, determine ifuis orthogonal to the subspace S. u =  2 3 1  , S = span  1 2 1  ,  3 1 1 

For Exercises 2728, determine ifuis orthogonal to the subspace S. u = 0 1 0 1 , S = span 1 1 0 1 , 1 0 1 0 , 1 1 1 1

For Exercises 2932, find a basis for S for the subspace S. = span  1 3 

For Exercises 2932, find a basis for S for the subspace S. S = span2 5 

For Exercises 2932, find a basis for S for the subspace S. = span  1 1 2 

For Exercises 2932, find a basis for S for the subspace S. S = span 1 2 1  ,  2 3 2 

For Exercises 3334, show that the given basis for S is orthogonal, and then write s as a linear combination of the basis vectors. = span 1 1 0  ,  1 1 4  , s =  1 2 2

For Exercises 3334, show that the given basis for S is orthogonal, and then write s as a linear combination of the basis vectors. S = span 1 0 1 1 , 3 3 1 4 , 2 3 1 1 , s = 1 0 0 1

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. Two vectors u and v such that u v = 12.

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. A vector that is orthogonal to both  1 0 and  0 1

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. A vector u such that u = 1 and u is orthogonal to  2 1

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. An orthogonal basis for R2 that includes 3 4

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. Two linearly independent vectors that are both orthogonal to  2 0 1  .

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. Two vectors in R2 that are orthogonal but do not span R2.

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. A subspace S of R3 that has dim(S) = 2.

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. Two vectors u and v such that u, v, and 1 1 1  form an orthogonal set.

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. Three vectors in R3 that form an orthogonal set but not an orthogonal basis.

FIND AN EXAMPLE For Exercises 3544, find an example that meets the given specifications. A subspace S of R4 such that dim(S) = dim(S) = 2.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If u v = 3, then the distance between 2u and 2v is 12.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If u and v have nonnegative entries, then u v 0.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. u + v=u+v for all u and v in Rn.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. Suppose that {s1, s2, s3} is an orthogonal set and that c1, c2, and c3 are scalars. Then {c1s1, c2s2, c3s3} is also an orthogonal set.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If S is a one-dimensional subspace of R2, then so is S.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If A is an n n matrix and u is in Rn, then uAu.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If u1 u2 = 0 and u2 u3 = 0, then u1 u3 = 0.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If A is an n n matrix with orthogonal columns, then AT A is a diagonal matrix.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If u and v are orthogonal, then the distance between u and v is u2 + v2.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If u v=u + v, then u and v are orthogonal.

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. If A =  a1 a2  and S = span{a1, a2}, then S = null(A).

TRUE OR FALSE For Exercises 4556, determine if the statement is true or false, and justify your answer. Even if S is merely a nonempty subset of Rn, the orthogonal complement S is still a subspace.

Prove that Theorem 8.9 is true even if the set S = {s1, ... , sk } only spans the subspace S instead of being a basis for S.

Prove that the zero vector 0 in Rn is orthogonal to all vectors in Rn.

Prove that the standard basis {e1, ... , en} of Rn is an orthogonal basis.

Prove that if u1 and u2 are both orthogonal to v, then so is u1 + u2.

Prove that if c1 and c2 are scalars and u1 and u2 are vectors, then (c1u1) (c2u2) = c1c2(u1 u2).

Let u, v, and w be in Rn, and let c be a scalar. Prove each part of Theorem 8.2. (a) u v = v u (b) (u + v) w = u w + v w (c) (cu) v = u (cv) = c(u v) (d) u u 0, and u u = 0 only when u = 0

Use the properties of Theorem 8.2 to prove 1, which says that (c1u1 ++ ckuk ) w = c1(u1 w) ++ ck (uk w)

Prove 2, that cx=|c|x for a scalar c and vector x.

Prove that if u = 0 and v = 1 u u, then v = 1.

Let u be a vector in Rn, and then define Tu : Rn R by Tu(v) = u v. Show that Tu is a linear transformation.

Let S be a subspace. Prove that S S = {0}.

Prove that u + v2 + u v2 = 2  u2 + v2 

For a matrix A, show that  col(A)  = Null(AT ).

Prove that if S is a subspace, then S =  S.

Let u and v be in Rn and A be an n n matrix. (a) Explain why u v and uT v are essentially the same. (b) Show that (Au) v = u  AT v 

In this problem we show that if u and v are perpendicular vectors in R2, then u v = 0. The method of proof is different from the one given at the start of the subsection Orthogonal Vectors. (a) Given nonzero vectors u and v orthogonal in R2, show that u, v, and uv are the lengths of the sides of a right triangle. (b) Use the version of the Pythagorean Theorem you learned in high school geometry to show that u2 + v2 = u v2 (c) Write the equation in (b) in terms of dot products and then simplify to show that u v = 0.

C For Exercises 7374, let u1 = 3 1 5 0 2 , u2 = 7 4 0 2 8 , u3 = 0 3 4 4 3 Compute each of the following: (a) u2 u3 (b) u1 (c) 2u1 + 5u3 (d) 3u1 4u2 u3 74. C

C For Exercises 7374, let u1 = 3 1 5 0 2 , u2 = 7 4 0 2 8 , u3 = 0 3 4 4 3 Compute each of the following: (a) u2 u1 (b) u3 (c) 3u2 + 4u3 (d) -2u1 5u2 3u3

C For Exercises 7576, find a basis for S S = span 2 1 3 5 , 0 1 7 4

C For Exercises 7576, find a basis for S S = span 6 0 2 5 1 , 5 3 0 8 6