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Complex Vector Spaces

A vector space is a set of objects, called vectors, along with two operations, called addition and scalar multiplication, such that the sum of any two vectors and the scalar multiple of any vector is also a vector. The complex numbers form a vector space. Those would be the numbers of the form a+bi where a and b are real numbers. The sum of two complex numbers is defined by (a+bi)+(c+di)= (a+c)+(b+d)i and the scalar multiple of a complex number is defined by k(a+bi)=(ka)+(kb)i. The elements of any vector space satisfy the commutative and associative properties and every vector space has an identity element. Each element then has an additive inverse meaning their sum is the identity. A set of vectors v1,v2,v3,.....,vn is defined to be linearly independent if c1v1+c2v2+c3v3+.....+cnvn=0 implies c1=c2=c3=.....cn=0. An example of a set of linearly independent complex numbers is v1=1+i and v2=2-i. if c1v1+c2v2=0, then (c1+c1i)+(2c2-c2i)=0 which results in (c1+2c2)+(c1-c2)i=0. This equation can only be true if c1=c2=0. The two vectors v1=1+0i and v2=0+1i are linearly independent and form a basis for the entire vector space of complex numbers. This is because every vector can be expressed as linear combination of these two. For instance, the vector -3-7i can be written as -3v1+-7v2. The vectors v1 and v2 are said to span the vector space of complex numbers. Let z1=a+bi and z2=c+di. Using the definition of addition above, let s=(a+c)+(b+d)i.  Now let w1 be the vector from z1 pointing to s and w2 be the vector pointing from z2 to s. Then w1=(a+c-a)+(b+d-b)i or w1=c+di and w2=(a+c-c)+(b+d-d)i or w2=a+bi. It is apparent that w1=z2 and w2=z1 and therefore the four vectors make a quadrilateral with congruent opposite sides, otherwise known as a parallelogram.