8 1. Vecto r Spaces EXAMPLES: a. Solution-set of a system of homogeneous linear equations. Here Y — ¥n. Give n th e scalar s a t •, 1 i k, 1 j n, w e consider th e solution-se t o f th e syste m o f k homogeneous linea r equations (1.2.4) Xtf* 7 - = 0 , * =!,•••*. 7=1 This is the set of all n-tuples, [x 1? ... ,xn] G F" (thought of as vec- tors), for which all k equations are satisfied. I f both [x x ,... ,x n ] and bi ^ • • • ?}«] a r e solution s of (1.2.4), and a andZ? are scalars, then for each /, n n n L a iMxj+byj) = a L fl iA-+fo L w = °- 7=1 7= 1 7= 1 It follows that the solution-set of (1.2.4) is a subspace of ¥ n . b. I n F[JC] , th e spac e ¥ N [x] o f polynomial s Y^ a n^ o f degre e N. While F[JC] is an algebra, ¥N[x] is not an algebra why? c. I n the space C°° (R) of all infinitely differentiable, real-valued func- tions / o n R wit h the standard operations, the set of functions / that satisfy th e differential equatio n f'"(x)-5f"(x) + 2f'(x)-f(x) = 0. If we consider complex-valued solutions , then the field of scalars can be R or C. d. Subspace s of j^(n) : The set of diagonal matrices—the nxn matrice s with zero entries off the main diagonal, i.e., a- = 0 for / =£ j. The se t of symmetric matrices—the nxn matrice s whos e entrie s satisfy a tj = a jt .

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