6. Suppose V is a non-trivial complex inner product space. Show that the set of self- adjoint operators on V is NOT a subspace of L(V).

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.

Prove that in a given vector space V, the additive inverse of a vector is unique.

Prove that if A is similar to B and A is diagonalizable, then B is diagonalizable.

Find a basis for R2 that includes the vector (2,2).

Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.

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.