Subsection VSP Vector Space Properties

With definitions of vector addition and scalar multiplication we can state, and prove, several properties of each operation, and some properties that involve their interplay. We now collect ten of them here for later reference.

Theorem VSPCV Vector Space Properties of Column Vectors

Suppose that $\complex{m}$ is the set of column vectors of size $m$ (Definition VSCV) with addition and scalar multiplication as defined in Definition CVA and Definition CVSM. Then

  • ACC Additive Closure, Column Vectors
    If $\vect{u},\,\vect{v}\in\complex{m}$, then $\vect{u}+\vect{v}\in\complex{m}$.

  • SCC Scalar Closure, Column Vectors
    If $\alpha\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $\alpha\vect{u}\in\complex{m}$.

  • CC Commutativity, Column Vectors
    If $\vect{u},\,\vect{v}\in\complex{m}$, then $\vect{u}+\vect{v}=\vect{v}+\vect{u}$.

  • AAC Additive Associativity, Column Vectors
    If $\vect{u},\,\vect{v},\,\vect{w}\in\complex{m}$, then $\vect{u}+\left(\vect{v}+\vect{w}\right)=\left(\vect{u}+\vect{v}\right)+\vect{w}$.

  • ZC Zero Vector, Column Vectors
    There is a vector, $\zerovector$, called the zero vector, such that $\vect{u}+\zerovector=\vect{u}$ for all $\vect{u}\in\complex{m}$.

  • AIC Additive Inverses, Column Vectors
    If $\vect{u}\in\complex{m}$, then there exists a vector $\vect{-u}\in\complex{m}$ so that $\vect{u}+ (\vect{-u})=\zerovector$.

  • SMAC Scalar Multiplication Associativity, Column Vectors
    If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $\alpha(\beta\vect{u})=(\alpha\beta)\vect{u}$.

  • DVAC Distributivity across Vector Addition, Column Vectors
    If $\alpha\in\complex{\null}$ and $\vect{u},\,\vect{v}\in\complex{m}$, then $\alpha(\vect{u}+\vect{v})=\alpha\vect{u}+\alpha\vect{v}$.

  • DSAC Distributivity across Scalar Addition, Column Vectors
    If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $(\alpha+\beta)\vect{u}=\alpha\vect{u}+\beta\vect{u}$.

  • OC One, Column Vectors
    If $\vect{u}\in\complex{m}$, then $1\vect{u}=\vect{u}$.

Proof

Many of the conclusions of our theorems can be characterized as “identities,” especially when we are establishing basic properties of operations such as those in this section. Most of the properties listed in Theorem VSPCV are examples. So some advice about the style we use for proving identities is appropriate right now. Have a look at Proof Technique PI.

Be careful with the notion of the vector $\vect{-u}$. This is a vector that we add to $\vect{u}$ so that the result is the particular vector $\zerovector$. This is basically a property of vector addition. It happens that we can compute $\vect{-u}$ using the other operation, scalar multiplication. We can prove this directly by writing that \begin{equation*} \vectorentry{\vect{-u}}{i} =-\vectorentry{\vect{u}}{i} =(-1)\vectorentry{\vect{u}}{i} =\vectorentry{(-1)\vect{u}}{i} \end{equation*} We will see later how to derive this property as a consequence of several of the ten properties listed in Theorem VSPCV.

Similarly, we will often write something you would immediately recognize as “vector subtraction.” This could be placed on a firm theoretical foundation — as you can do yourself with Exercise VO.T30.

A final note. Property AAC implies that we do not have to be careful about how we “parenthesize” the addition of vectors. In other words, there is nothing to be gained by writing $\left(\vect{u}+\vect{v}\right)+\left(\vect{w}+\left(\vect{x}+\vect{y}\right)\right)$ rather than $\vect{u}+\vect{v}+\vect{w}+\vect{x}+\vect{y}$, since we get the same result no matter which order we choose to perform the four additions. So we will not be careful about using parentheses this way.

Sage ANC A Note on Coercion
(in context)