Consider the inner product space (R3, (.,.)), where (u, v) = u Aʊ, Vu, v € R³, and Let and define the set 1 21 A = 2 5 1 1 2 3 TH I · ( }) S = {R³ | is orthogonal to }, which can be shown to be a subspace of R3. Find a basis for S.

Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.

Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.

Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.

Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.

Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.

Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.

Find the bases for the four fundamental subspaces of the matrix. A=[010030101].

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba]}

Find a basis for the subspace of R' that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 1, O), v3 = (2, 1, 0), v4 = (0, -2, 0) O v and v2 form a basis for span {v1, V2, V3, V43. V, and v3 form a basis for span {v1, V2, V3, V4}. v2 and v4 form a basis for span {v1, V2, V3, V4}. O v, and v4 form a basis for span {v1, V2, V3, V4}. O v2 and v3 form a basis for span{v1, v2, v3, V4}. O v3 and v4 form a basis for span {v1, V2, v3, V4}. O All of the above are correct.