Math - Vectors - Vector spaces and some vector properties

According to Kaplan [1], a vector space, \( \left( v_1, \ldots, v_n \right) = V^{\; n} \), contains the following properties:

\begin{align} \mathbf{I.} &= \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \\

\mathbf{II.} &= \left( \mathbf{u} + \mathbf{v} \right) + \mathbf{w} = \mathbf{u} + \left( \mathbf{v} + \mathbf{w} \right) \\

\mathbf{III.} &= h \left( \mathbf{u} + \mathbf{v} \right) = h \mathbf{u} + h \mathbf{v} \\

\mathbf{IV.}&= \left( a + b \right) \mathbf{u} = a \mathbf{u} + b \mathbf{u} \\

\mathbf{V.} &= \left( a b \right) \mathbf{u} = a \left( b \mathbf{u} \right) \\

\mathbf{VI.} &= 1 \mathbf{u} = \mathbf{u} \\

\mathbf{VII.} &= 0 \mathbf{u} = \mathbf{0} \\

\mathbf{VIII.} &= \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \\

\mathbf{IX.} &= \left( \mathbf{u} + \mathbf{v} \right) \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w} \\

\mathbf{X.} &= \left( a \mathbf{u} \right) \cdot \mathbf{v} = a \left( \mathbf{u} \cdot \mathbf{v} \right) \\

\mathbf{XI.} &= \mathbf{u} \cdot \mathbf{u} \ge 0 \\

\mathbf{XII.}  &= \mathbf{u} \cdot \mathbf{u} = 0 \; \; \text{if and only if} \; \; \mathbf{u} = 0 \end{align}

The first property in the above list is known as the commutative property or law which is true for vector addition [2]:

\[ \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \]

or with scalars [3]

\[ m \mathbf{u} = \mathbf{u}m \]

Note:  I think there might be an error in Tai [4].  He states that the associative law is

\[  \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \]

or

\[ \mathbf{A} - \mathbf{B} = -\mathbf{B} + \mathbf{A} \]

when I think he meant commutative.

The second property in the above list is known as the associative property or law which is also true for vector addition [2, 3]:

\[ \left( \mathbf{u} + \mathbf{v} \right) + \mathbf{w} = \mathbf{u} + \left( \mathbf{v} + \mathbf{w} \right) \]

or with scalars [3]

\[ m \left( n \mathbf{u} \right) = \left (mn \right) \mathbf{u}m \]

The third property in the above list is known as the distributive property or law which is also true for vector addition [3]:

\[  \left( m + n \right) \mathbf{u} = m \mathbf{u} + n \mathbf{u} \]

or

\[  m \left(  \mathbf{u} +  \mathbf{v} \right) = m \mathbf{u} + m \mathbf{v} \]

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References:

[1] W. Kaplan. Advanced Calculus, 5th ed. Addison-Wesley. 2002

[2] A. I. Borisenko and I. E. Tarapov. Vector and Tensor Analysis with Applications, (translated by R. A. Silverman). Dover Publications Inc., Mineola, NY. 1979 (originally published in 1968 by Prentice-Hall, Inc.

[3] T. C. Papanastasiou, G. C. Georgiou, & A. N. Alexandrou. Viscous Fluid Flow. CRC Press. Boca Raton, FL. 2000

[4] C.-T. Tai. General Vector and Dyadic Analysis: Applied Mathematics in Field Theory, 2nd ed. Wiley-IEEE Press, New York, NY. 1997.