Arithmetic Theorems

Section 2.2 Arithmetic Theorems

We will

In this section we want to connect properties of vectors to similar looking properties of real numbers we already know.

Select a name to match each property. Be careful to distinguish between scalar and vector. For example,

\vec{u} + \vec{v} = \vec{v} + \vec{u}

is vector additive commutativity.

Select a name for each property. Be careful to distinguish between scalar and matrix.

Select a name for each property. Be careful to distinguish between scalar, vector, and matrix.

Select a name for each property. No two have the exact same description: be precise! The last item will be explained in a future lesson.

c (\vec{u} + \vec{v}) = c \vec{u} + c \vec{v}

\begin{aligned} c (\vec{u} + \vec{v}) = & c ([u_{1}, u_{2}, \dots, u_{n}] + [v_{1}, v_{2}, \dots, v_{n}]) \\ = & c ([u_{1} + v_{1}, u_{2} + v_{2}, \dots, u_{n} + v_{n}]) \\ definition of vector addition \\ = & [c (u_{1} + v_{1}), c (u_{2} + v_{2}), \dots, c (u_{n} + v_{n})] \\ definition of scalar multiplication \\ = & [c u_{1} + c v_{1}, c u_{2} + c v_{2}, \dots, c u_{n} + c v_{n}] \\ property of real arithmetic \\ = & [c u_{1}, c u_{2}, \dots, c u_{n}] + [c v_{1}, c v_{2}, \dots, c v_{n}] \\ definition of vector addition \\ = & c [u_{1}, u_{2}, \dots, u_{n}] + c [v_{1}, v_{2}, \dots, v_{n}] \\ definition of scalar multiplication \\ = & c \vec{u} + c \vec{v} . \end{aligned}

Prove the following by doing the algebra steps.