A D,C are three linear transformations on a vector space V(F) such that AB = CA = I, then prove that A is invertible and A =B = C.

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 f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

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.

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.

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.

Prove that in a given vector space V, the zero vector is unique.

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

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