Section 1.2: Vector Spaces

Annotations:

• Vector Space over a field F: A set on which two operations (addition and scalar multiplication) are defined so that all x,y in V are closed under these operations. In addition the following conditions must hold: VS 1 [Commutativity of +] - For all x,y in V, x+y = y+x VS 2 [Associativity of +] - For all x,y,z in V, (x+y)+z = x+(y+z) VS 3 [+ Identity] - There exists an element in V (denoted by 0) s.t. x+0 = x, for all x in V VS 4 [+ Inverse] - For all x in V, there exists an element y in V s.t. x+y = 0 VS 5 [Scalar Identity] - For all x in V, 1x = x VS 6 [Associativity of Scalar] - For all a, b in F and x in V, (ab)x = a(bx) VS 7 [Distributivity of Scalar] - For all a in F and x,y in V, a(x+y) = ax + ay VS 8 [Distributivity of Vector] - For all a,b in F and x in V, (a+b)x = ax + bx

1. Theorem 1.1 (Cancellation Law for Vectors)

   Annotations:

   • Theorem 1.1 [Cancellation Law for Vector Addition]: If x,y,z are vectors in V s.t. x+z = y+z --> x=y
2. Corollary (Zero Vector)

   Annotations:

   • Corollary [Zero Vector] - The 0 vector, s.t. x+0=0, is unique
3. Corollary (Inverse Vectors)

   Annotations:

   • Corollary [Inverse Vectors]: The inverse vector, y, s.t. x+y=0, is unique 
4. Theorem 1.2 (Properties of a Vector Space)

   Annotations:

   • Theorem 1.2 [Properties of a Vector Space] - a) 0x = 0 for all x in V b) -a(x) = -(ax) = a(-x) for all a in F, x in V c) a0 = 0 for all a in F
5. Zero Vector Space

   Annotations:

   • V = {0} is the zero vector space
6. Even Function/Odd Function

   Annotations:

   • A real-valued function, f, is called an even function if f(-t) = f(t) A real-valued function, f, is called an odd function if f(-t) = -f(t)