1.2. Vecto r spaces 9 The set of skew-symmetric matrices—the nxn matrice s whose en- tries satisfy a t - = —a-t. The set of lower triangular matrices—the nxn matrice s with zero entries above the main diagonal, i.e., a- = 0 for / j. Similarly, the set of upper triangular matrices, i.e., those for which a- = Ofor / /. I J J Remark: I n view o f the isomorphis m betwee n M{n) and ¥ n , these subspaces can be viewed as special cases of example a. e. Intersection of subspaces: If Wj are subspaces of a space y, j G /, (the index set J can be finite or infinite), then f| W}is a subspace o f r . /. The sum of subspaces: If Wj, j G /, are subspaces of y, thei r sum is the set: where the union extends over the collection of finite subsets of /. Don't confuse th e sum of subspaces wit h the union of subspaces, which is seldom a subspace, see exercises exl.2.5 and exl.2.4. g. The span of a subset: Give n E c Y, a linear combination of ele- ments of E is a finite sum of the form Lfl/V/ ^ G F , V J G £. Th e set of all the linear combinations of elements of £ is a subspace of y, calle d the span of E and denoted by span [E]. 1.2.4 Quotien t spaces. A subspace W of a vector space ^ define s an equivalence relation (see the appendix, section A.l) in Y: (1.2.5) x = y (mo&W) i f x-yeW. In order to establish that this is indeed an equivalence relation, we need to verify that it is reflexive, symmetric, and transitive: a. reflexive: x = JC, since x — x = 0e W,

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