Problem 1
For vector $\boldsymbol{x}$ in $\mathbb{R}^3$,
define $f(\boldsymbol{x}) = \boldsymbol{w} \cdot (2 \cdot \boldsymbol{x} + \boldsymbol{v})$,
where $\boldsymbol{w} = [1\ -1/2\ 3]$ and $\boldsymbol{v} = [0\ 1\ -1]$.
If $\boldsymbol{x} = [-4\ 3\ 5]$, what is $f(\boldsymbol{x})$?
Problem 2
For vector $\boldsymbol{x}$ in $\mathbb{Z}_2^3$ define
$g(\boldsymbol{x}) = \boldsymbol{w} \cdot \boldsymbol{x} +
\boldsymbol{v}$ , note the difference from first problem!.
where $\boldsymbol{w} = [1\ 0\ 1]$ and $\boldsymbol{v} = [1\ 1\ 0]$.
If $\boldsymbol{x} = [0\ 1\ 1]$, what is $g(\boldsymbol{x})$?
What if we define $g(\boldsymbol{x}) = \boldsymbol{w} \cdot (\boldsymbol{x} +
\boldsymbol{v})$ instead?
Proving properties of vector operations
Prove one of the following at the whiteboard. Make sure all are
covered by at least one group!
- vector addition is commutative and associative
- dot product is commutative, but dot product is *not*
associative!
-
$a\cdot\boldsymbol{u} + a\cdot\boldsymbol{v}
= a\cdot(\boldsymbol{u} + \boldsymbol{v})$ where $a$ is scalar
and $\boldsymbol{u}$ and $\boldsymbol{v}$ are vectors
- $(a\cdot b)\cdot \boldsymbol{u} = a\cdot (b\cdot
\boldsymbol{u})$
where $a$ and $b$ are scalars and
$\boldsymbol{u}$ is a vector
- $a\cdot (\boldsymbol{u}\cdot \boldsymbol{v}) =
(a\cdot \boldsymbol{u})\cdot \boldsymbol{v} =
\boldsymbol{u}\cdot (a\cdot \boldsymbol{v})$
where $a$ is scalar and
$\boldsymbol{u}$ and $\boldsymbol{v}$ are vectors
- $(\boldsymbol{u} + \boldsymbol{v})\cdot \boldsymbol{w} =
\boldsymbol{u}\cdot \boldsymbol{w} + \boldsymbol{v}\cdot \boldsymbol{w}$,
where $\boldsymbol{u}$, $\boldsymbol{v}$ and $\boldsymbol{w}$ are vectors
Note: If you finish proving one item, choose another and
keep going!