1.2. Vecto r spaces 5 a. F n , the space of all F-valued ^-tuples [a x ,..., a n ] with addition and multiplication by scalars defined by [av...,an] + [bv...,bn] = [a x +bv... ,a n + bn], c[aX)...,an} = [ca v ...,can]. We may write the n-tuples as rows, as we did here, or as columns. If we want to specify tha t the vectors are written as columns or as rows, we write F" or F", respectively. b. JK(n, m F), the space of all F-valued nxm matrices that is, arrays fall '• a lm\ \a2l .. . a 2m A- = with entries from F . W e sometimes write A = [a-] when the di- mensions of the matrix are assumed known, to save space. The additio n an d multiplicatio n b y scalar s ar e agai n don e entr y by entry. As a vector space, J%(n,m\¥) i s virtually identical with TCVW ! We write J£{n\¥) instea d o f JZ(n,n\W), an d i f th e underlyin g field is either known implicitly, or assumed explicitly, then we of - ten write simply jti£(n,ni) or M(n), a s the case may be. c. F[x] , the space 1 o f al l polynomials £a w ^ wit h coefficients fro m F. Additio n and multiplication by scalars are defined formally ei- ther as the standard addition and multiplication of functions (of the variable x ), or by adding (and multiplying by scalars) the corre- sponding coefficients. Th e two ways define the same operations. More generally, the set ¥[x v ... ,x k ] o f al l polynomials i n k vari- ables over F is a vector space (in fact—an algebra). l ¥[x] i s a n algebr a ove r F , i.e. , a vector spac e wit h th e additiona l operatio n o f multiplication. Se e A.5.6.
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