We proceed now to derive certain elementary properties of, and relations among, linear transformations on a vector space. More particularly, we shall indicate several ways of making new transformations out of old ones; we shall generally be satisfied with giving the definition of the new transformations and we shall omit the proof of linearity.

If \(A\) and \(B\) are linear transformations, we define their sum, \(S=A+B\) , by the equation \(S x=A x+B x\) (for every \(x\) ). We observe that the commutativity and associativity of addition in \(\mathcal{V}\) imply immediately that the addition of linear transformations is commutative and associative. Much more than this is true. If we consider the sum of any linear transformation \(A\) and the linear transformation \(0\) (defined in the preceding section), we see that \(A+0=A\) . If, for each \(A\) , we denote by \(-A\) the transformation defined by \((-A) x=-(A x)\) , we see that \(A+(-A)=0\) , and that the transformation \(-A\) , so defined, is the only linear transformation \(B\) with the property that \(A+B=0\) . To sum up: the properties of a vector space, described in the axioms (A) of Section: Vector spaces , appear again in the set of all linear transformations on the space; the set of all linear transformations is an abelian group with respect to the operation of addition.

We continue in the same spirit. By now it will not surprise anybody if the axioms (B) and (C) of vector spaces are also satisfied by the set of all linear transformations. They are. For any \(A\) , and any scalar \(\alpha\) , we define the product \(\alpha A\) by the equation \((\alpha A) x=\alpha(A x)\) . Axioms (B) and (C) are immediately verified; we sum up as follows.

Theorem 1. The set of all linear transformations on a vector space is itself a vector space.

We shall usually ignore this theorem; the reason is that we can say much more about linear transformations, and the mere fact that they form a vector space is used only very rarely. The “much more” that we can say is that there exists for linear transformations a more or less decent definition of multiplication, which we discuss in the next section.

EXERCISES

Exercise 1. Prove that each of the correspondences described below is a linear transformation.

  1. \(\mathcal{V}\) is the set \(\mathbb{C}\) of complex numbers regarded as a real vector space; \(A x\) is the complex conjugate of \(x\) .
  2. \(\mathcal{V}\) is \(\mathcal{P}\) ; if \(x\) is a polynomial, then \((A x)(t)=x(t+1)-x(t)\) .
  3. \(\mathcal{V}\) is the \(k\) -fold tensor product of a vector space with itself; \(A\) is such that \[A(x_{1} \otimes \cdots \otimes x_{k})=x_{\pi(1)} \otimes \cdots \otimes x_{\pi(k)},\] where \(\pi\) is a permutation of \(\{1, \ldots, k\}\) .
  4. \(\mathcal{V}\) is the set of all \(k\) -linear forms on a vector space; \[(A w)(x_{1}, \ldots, x_{k})=w(x_{\pi(1)}\ldots, x_{\pi(k)}),\] where \(\pi\) is a permutation of \(\{1, \ldots, k\}\) .
  5. \(\mathcal{V}\) is the set of all \(k\) -linear forms on a vector space; if \(w\) is in \(\mathcal{V}\) , then \(A w=\sum \pi w\) , where the summation is extended over all permutations \(\pi\) in \(\mathcal{S}_{k}\) .
  6. Same as (e) except that \(A w=\sum(\operatorname{sgn} \pi)\, \pi w\) .

Exercise 2. Prove that if \(\mathcal{V}\) is a finite-dimensional vector space, then the space of all linear transformations on \(\mathcal{V}\) is finite-dimensional, and find its dimension.

Exercise 3. The concept of a "linear transformation," as defined in the text, is too special for some purposes. According to a more general definition, a linear transformation from a vector space \(\mathcal{U}\) to a vector space \(\mathcal{V}\) over the same field is a correspondence \(A\) that assigns to every vector \(x\) in \(\mathcal{U}\) a vector \(A x\) in \(\mathcal{V}\) so that \[A(\alpha x+\beta y)=\alpha A x+\beta A y\] Prove that each of the correspondences described below is a linear transformation in this generalized sense.

  1. \(\mathcal{V}\) is the field of scalars of \(\mathcal{U}\) ; \(A\) is a linear functional on \(\mathcal{U}\) .
  2. \(\mathcal{U}\) is the direct sum of \(\mathcal{V}\) with some other space; \(A\) maps each pair in \(\mathcal{U}\) onto its first coordinate.
  3. \(\mathcal{V}\) is the quotient of \(\mathcal{U}\) modulo a subspace; \(A\) maps each vector in \(\mathcal{U}\) onto the coset it determines.
  4. Let \(w\) be a bilinear functional on a direct sum \(\mathcal{U} \oplus \mathcal{V}_{0}\) . Let \(\mathcal{V}\) be the dual of \(\mathcal{V}_{0}\) , and define \(A\) to be the correspondence that assigns to each \(x_{0}\) in \(\mathcal{U}\) the linear functional on \(\mathcal{V}_{0}\) obtained from \(w\) by setting its first argument equal to \(x_{0}\) .

Exercise 4. 

  1. Suppose that \(\mathcal{U}\) and \(\mathcal{V}\) are vector spaces over the same field. If \(A\) and \(B\) are linear transformations from \(\mathcal{U}\) to \(\mathcal{V}\) , if \(\alpha\) and \(\beta\) are scalars, and if \[C x=\alpha A x+\beta B x\] for each \(x\) in \(\mathcal{U}\) , then \(C\) is a linear transformation from \(\mathcal{U}\) to \(\mathcal{V}\) .
  2. If we write, by definition, \(C=\alpha A+\beta B\) , then the set of all linear transformations from \(\mathcal{U}\) to \(\mathcal{V}\) becomes a vector space with respect to this definition of the linear operations.
  3. Prove that if \(\mathcal{U}\) and \(\mathcal{V}\) are finite-dimensional, then so is the space of all linear transformations from \(\mathcal{U}\) to \(\mathcal{V}\) , and find its dimension.

Exercise 5. Suppose that \(\mathcal{M}\) is an \(m\) -dimensional subspace of an \(n\) -dimensional vector space \(\mathcal{V}\) . Prove that the set of those linear transformations \(A\) on \(\mathcal{V}\) for which \(A x=0\) whenever \(x\) is in \(\mathcal{M}\) is a subspace of the set of all linear transformations on \(\mathcal{V}\) , and find the dimension of that subspace.