2 1. Preliminaries • V is an abelian group under vector addition i.e., vector addition is commuta- tive and associative, there is a zero vector which is the additive identity, and every vector has an additive inverse. • Scalar multiplication satisfies the following properties: – For any a in F and x, y in V, we have a(x + y) = ax + ay. – For any a, b in F and x in V, we have (a + b)x = ax + bx. – For any a, b in F and x in V, we have (ab)x = a(bx). – For any x in V we have 1x = x. We use bold face letters to denote vectors. The term scalars refers to elements in F. It can be shown that for any vector x the product 0(x) is the zero vector, 0. The most basic example of a vector space is Fn, the set of all ordered n-tuples, x = (x1,...,xn), where the xi’s are elements of F, with vector addition and scalar multiplication defined in the usual coordinate-wise manner: • (x1,...,xn) + (y1,...,yn) = (x1 + y1,...,xn + yn). • a(x1,...,xn) = (ax1,...,axn). Although we write x = (x1,...,xn) in row form for typographical convenience, we generally regard vectors in Fn as column vectors, in order to conform with our notation for the matrix of a linear transformation, presented later in this chapter. A nonempty subset U of V is a subspace of V if U is itself a vector space over F, using the same operations as in V. Equivalently, a nonempty subset U is a subspace of V if U is closed under vector addition and scalar multiplication. If U and W are subspaces, then the intersection U ∩ W is a subspace, but, unless either U ⊆ W or W ⊆ U, the union U ∪ W is not a subspace. However, the sum, U + W = {u + w | u ∈ U and w ∈ W}, is a subspace. If for each element z of U + W there is exactly one choice of u ∈ U and w ∈ W such that z = u + w, then we say the sum U + W is a direct sum and write U ⊕ W. Equivalently, the sum U + W is a direct sum if and only if U ∩ W = {0}. A vector y is a linear combination of the vectors x1,..., xn if y = ∑n i=1 aixi, for some a1,...,an in F. The set of all possible linear combinations of x1,..., xn is the span of x1,..., xn, denoted span[x1,..., xn]. For an infinite set S of vectors, span[S] is the set of all possible linear combinations of a finite number of vectors in S. The span of the empty set is defined to be the zero subspace, {0}. For any subset S of vectors in V, the span of S is a subspace of V, called the subspace spanned by S and denoted as span[S]. If U is a subspace of V and U= span[S], we say S is a spanning set for U, or S spans U. The vectors x1,..., xn are said to be linearly dependent if there exist scalars a1,...,an, not all zero, such that ∑n i=1 aixi = 0. Equivalently, x1,..., xn are linearly dependent if one of the vectors xi is a linear combination of the other vectors in the list. An infinite set S of vectors is linearly dependent if it has a finite subset which is linearly dependent. The vectors x1,..., xn are linearly independent if they are not linearly dependent, i.e., if the only choice of scalars for which ∑n i=1 aixi = 0 is

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