4. Suppose TЄ L(V) and U is a subspace of V. Prove that U is invariant under T if and only if U is invariant under T*.

Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,

Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.

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.

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.