In most of what follows we shall view the notion of direct sum as defined for subspaces of a vector space \(\mathcal{V}\) ; this avoids the fuss with the identification convention of Section: Direct sums , and it turns out, incidentally, to be the more useful concept for our later work. We conclude, for the present, our study of direct sums, by observing the simple relation connecting dual spaces, annihilators, and direct sums. To emphasize our present view of direct summation, we return to the letters of our earlier notation.

Theorem 1. If \(\mathcal{M}\) and \(\mathcal{N}\) are subspaces of a vector space \(\mathcal{V}\) , and if \(\mathcal{V} = \mathcal{M} \oplus \mathcal{N}\) , then \(\mathcal{M}'\) is isomorphic to \(\mathcal{N}^0\) and \(\mathcal{N}'\) to \(\mathcal{M}^0\) , and \(\mathcal{V}' = \mathcal{M}^0 \oplus \mathcal{N}^0\) .

Proof. To simplify the notation we shall use, throughout this proof, \(x\) , \(x'\) , and \(x^0\) for elements of \(\mathcal{M}\) , \(\mathcal{M}'\) , and \(\mathcal{M}^0\) , respectively, and we reserve, similarly, the letters \(y\) for \(\mathcal{N}\) and \(z\) for \(\mathcal{V}\) . (This notation is not meant to suggest that there is any particular relation between, say, the vectors \(x\) in \(\mathcal{M}\) and the vectors \(x'\) in \(\mathcal{M}'\) .)

If \(z'\) belongs to both \(\mathcal{M}^0\) and \(\mathcal{N}^0\) , i.e., if \(z'(x) = z'(y) = 0\) for all \(x\) and \(y\) , then \(z'(z) = z'(x + y) = 0\) for all \(z\) ; this implies that \(\mathcal{M}^0\) and \(\mathcal{N}^0\) are disjoint. If, moreover, \(z'\) is any vector in \(\mathcal{V}'\) , and if \(z = x + y\) , we write \(x^0(z) = z'(y)\) and \(y^0(z) = z'(x)\) . It is easy to see that the functions \(x^0\) and \(y^0\) thus defined are linear functionals on \(\mathcal{V}\) (i.e., elements of \(\mathcal{V}'\) ) belonging to \(\mathcal{M}^0\) and \(\mathcal{N}^0\) respectively; since \(z' = x^0 + y^0\) , it follows that \(\mathcal{V}'\) is indeed the direct sum of \(\mathcal{M}^0\) and \(\mathcal{N}^0\) .

To establish the asserted isomorphisms, we make correspond to every \(x^0\) a \(y'\) in \(\mathcal{N}'\) defined by \(y'(y) = x^0(y)\) . We leave to the reader the routine verification that the correspondence \(x^0 \to y'\) is linear and one-to-one, and therefore an isomorphism between \(\mathcal{M}^0\) and \(\mathcal{N}'\) ; the corresponding result for \(\mathcal{N}^0\) and \(\mathcal{M}'\) follows from symmetry by interchanging \(x\) and \(y\) . (Observe that for finite-dimensional vector spaces the mere existence of an isomorphism between, say, \(\mathcal{M}^0\) and \(\mathcal{N}'\) is trivial from a dimension argument; indeed, the dimensions of both \(\mathcal{M}^0\) and \(\mathcal{N}'\) are equal to the dimension of \(\mathcal{N}\) .) ◻

We remark, concerning our entire presentation of the theory of direct sums, that there is nothing magic about the number two; we could have defined the direct sum of any finite number of vector spaces, and we could have proved the obvious analogues of all the theorems of the last three sections, with only the notation becoming more complicated. We serve warning that we shall use this remark later and treat the theorems it implies as if we had proved them.

EXERCISES

Exercise 1. Suppose that \(x\) , \(y\) , \(u\) , and \(v\) are vectors in \(\mathbb{C}^4\) ; let \(\mathcal{M}\) and \(\mathcal{N}\) be the subspaces of \(\mathbb{C}^4\) spanned by \(\{x, y\}\) and \(\{u, v\}\) respectively. In which of the following cases is it true that \(\mathbb{C}^4 = \mathcal{M} \oplus \mathcal{N}\) ?

  1. \(x = (1, 1, 0, 0)\) , \(y = (1, 0, 1, 0)\) , \(u = (0, 1, 0, 1)\) , \(v = (0, 0, 1, 1)\) .
  2. \(x = (-1, 1, 1, 0)\) , \(y = (0, 1, -1, 1)\) , \(u = (1, 0, 0, 0)\) , \(v = (0, 0, 0, 1)\) .
  3. \(x = (1, 0, 0, 1)\) , \(y = (0, 1, 1, 0)\) , \(u = (1, 0, 1, 0)\) , \(v = (0, 1, 0, 1)\) .

Exercise 2. If \(\mathcal{M}\) is the subspace consisting of all those vectors \((\xi_1, \ldots, \xi_n, \xi_{n+1}, \ldots, \xi_{2n})\) in \(\mathbb{C}^{2n}\) for which \(\xi_1 = \cdots = \xi_n = 0\) , and if \(\mathcal{N}\) is the subspace of all those vectors for which \(\xi_j = \xi_{n+j}\) , \(j = 1, \ldots, n\) , then \(\mathbb{C}^{2n} = \mathcal{M} \oplus \mathcal{N}\) .

Exercise 3. Construct three subspaces \(\mathcal{M}\) , \(\mathcal{N}_1\) , \(\mathcal{N}_2\) of a vector space \(\mathcal{V}\) so that \(\mathcal{M} \oplus \mathcal{N}_1 = \mathcal{M} \oplus \mathcal{N}_2 = \mathcal{V}\) but \(\mathcal{N}_1 \neq \mathcal{N}_2\) . (Note that this means that there is no cancellation law for direct sums.) What is the geometric picture corresponding to this situation?

Exercise 4. 

  1. If \(\mathcal{U}\) , \(\mathcal{V}\) , and \(\mathcal{W}\) are vector spaces, what is the relation between \(\mathcal{U} \oplus (\mathcal{V} \oplus \mathcal{W})\) and \((\mathcal{U} \oplus \mathcal{V}) \oplus \mathcal{W}\) (i.e., in what sense is the formation of direct sums an associative operation)?
  2. In what sense is the formation of direct sums commutative?

Exercise 5. 

  1. Three subspaces \(\mathcal{L}\) , \(\mathcal{M}\) , and \(\mathcal{N}\) of a vector space \(\mathcal{V}\) are called independent if each one is disjoint from the sum of the other two. Prove that a necessary and sufficient condition for \(\mathcal{V} = \mathcal{L} \oplus (\mathcal{M} \oplus \mathcal{N})\) (and also for \(\mathcal{V} = (\mathcal{L} \oplus \mathcal{M}) \oplus \mathcal{N}\) ) is that \(\mathcal{L}\) , \(\mathcal{M}\) , and \(\mathcal{N}\) be independent and that \(\mathcal{V} = \mathcal{L} + \mathcal{M} + \mathcal{N}\) . (The subspace \(\mathcal{L} + \mathcal{M} + \mathcal{N}\) is the set of all vectors of the form \(x + y + z\) , with \(x\) in \(\mathcal{L}\) , \(y\) in \(\mathcal{M}\) , and \(z\) in \(\mathcal{N}\) .)
  2. Give an example of three subspaces of a vector space \(\mathcal{V}\) , such that the sum of all three is \(\mathcal{V}\) , such that every two of the three are disjoint, but such that the three are not independent.
  3. Suppose that \(x\) , \(y\) , and \(z\) are elements of a vector space and that \(\mathcal{L}\) , \(\mathcal{M}\) , and \(\mathcal{N}\) are the subspaces spanned by \(x\) , \(y\) , and \(z\) , respectively. Prove that the vectors \(x\) , \(y\) , and \(z\) are linearly independent if and only if the subspaces \(\mathcal{L}\) , \(\mathcal{M}\) , and \(\mathcal{N}\) are independent.
  4. Prove that three finite-dimensional subspaces are independent if and only if the sum of their dimensions is equal to the dimension of their sum.
  5. Generalize the results (a)-(d) from three subspaces to any finite number.