# Lecture12 - .1 Vector spaces with complex scalars A complex...

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1 Lecture 12 12.1 Vector spaces with complex scalars A complex vector space is a set V of vectors together with vector addition w v in V and scalar multiplication v r in V by elements of C. Moreover, for all the vectors in V, and scalars in C, the following properties are satisfied: Properties of vector addition: A1: ( u + v ) + w = u + ( v + w ) an associatelaw A2: u + w = w + u a commutative law A3: 0 + v = v 0 as additive identity A4: v + (- v ) = 0 - v as additive inverse of v Properties of vector scalar multiplication: S1: r ( v + w ) = (r v ) + (r w ) a distributive law S2: (r + s) v =( r v ) + (s v ) a distributive law S3: r (s v ) = (rs) v an associative law S4: 1 v = v preservation of scale An example of complex vector space is the Euclidean space C n consisting of all vectors of the form   n u u u u ..., , , 2 1 , where i u C. Example: Consider the set V of all matrix of the form a a 0 0 where a is in C, with matrix addition and scalar multiplication. Determine whether V forms a complex vector space. Example: The set of m n matrices with complex entries and the operations of matrix addition and scalar multiplication is a complex vector space. Example: If f 1 (x) and f 2 (x) are real-valued functions of real variable x, then the expression f(x) = f 1 (x) + if 2 (x) is called a complex-valued function of the real variable x. The set of all f(x) with usual operations is a complex vector space.

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Lecture12 - .1 Vector spaces with complex scalars A complex...

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