Exercise 5 — Linear Algebra

Prove that the dimensionless products form a vector space under the $\vec{+}$ operation of multiplying two such products and the $\vec{\cdot}$ operation of raising such the product to the power of the scalar. (The vector arrows are a precaution against confusion.) That is, prove that, for any particular homogeneous system, this set of products of powers of $m_{1}$ , …, $m_{k}$

\{\hskip 1.00006ptm_{1}^{p_{1}}\dots m_{k}^{p_{k}}\mid\text{$p_{1}$, \ldots, $p_{k}$ satisfy the system}\hskip 1.00006pt\}

is a vector space under:

m_{1}^{p_{1}}\dots m_{k}^{p_{k}}\vec{+}m_{1}^{q_{1}}\dots m_{k}^{q_{k}}=m_{1}^{p_{1}+q_{1}}\dots m_{k}^{p_{k}+q_{k}}

and

r\vec{\cdot}(m_{1}^{p_{1}}\dots m_{k}^{p_{k}})=m_{1}^{rp_{1}}\dots m_{k}^{rp_{k}}

(assume that all variables represent real numbers).

Hefferon, Linear Algebra, Vector Spaces, Dimensional Analysis, Exercise 5

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