1.3. Linear Transformations 3 the trivial choice a1 = a2 = · · · = an = 0. An infinite set S of vectors is said to be linearly independent if every finite subset is linearly independent. We say a vector space V is finite dimensional if there is a finite set of vectors which spans V. Otherwise, V is infinite dimensional. For example, Fn is spanned by the unit coordinate vectors e1,..., en, where ei is the vector with a one in position i and zeroes elsewhere. So, Fn is finite dimensional. However, the set of all infinite sequences of elements of F, {(a1,a2,a3,... ) | ai F}, is infinite dimensional. In this book, we are concerned mainly with finite dimensional spaces. 1.2. Bases and Coordinates Let V be a finite dimensional vector space. Any linearly independent set of vectors which spans V is called a basis for V. It can be shown that every basis for V has the same number of elements. The number of elements in a basis of V is the dimension of V. For example, {e1,..., en} is a basis for Fn, and Fn is n-dimensional. If V is n-dimensional, then any spanning set for V has at least n elements, and any linearly independent subset of V has at most n elements. If U is a subspace of the n-dimensional vector space V, then dim(U) n, and dim(U) = n if and only if U = V. If k n and v1,..., vk are linearly independent, then there exist vectors vk+1,..., vn such that {v1,..., vk, vk+1,..., vn} is a basis for V i.e., any linearly independent set can be extended to a basis. A one-dimensional subspace is called a line we denote the line spanned by x as tx, representing {tx | t F}. Two linearly independent vectors, u and w, span a plane, which is a two-dimensional subspace. We use su + tw to denote the plane span[u, w]. A basis is the key tool for showing that any n-dimensional vector space over F may be regarded as the space Fn. Theorem 1.1. The set B = {b1,..., bn} is a basis for V if and only if each vector v V can be expressed in exactly one way as a linear combination of the vectors b1,..., bn i.e., for each vector v in V, there is a unique choice of coefficients v1,...,vn such that v = ∑n i=1 vibi. If B = {b1,..., bn} is a basis for V and v = ∑n i=1 vibi, then we call the co- efficients v1,...,vn the coordinates of v with respect to the B-basis and write [v] B = (v1,...,vn). Observe that for any scalars a, b in F, and vectors v, w in V, we have a[v]B + b[w]B = [av + bw]B. In the vector space Fn, when we write x = (x1,...,xn), the xi’s are the coordinates of x with respect to the standard basis {e1,..., en}. 1.3. Linear Transformations Let V and W be vector spaces over the field F. A function T : V W is called a linear transformation if T (ax + by) = aT (x) + bT (y), for all scalars a, b in F, and all vectors x, y in V. The null space or kernel of T is ker(T ) = {x V | T (x) = 0}.
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