Chapter 3.CE, Problem 18CE

In exercises 13-23, give a brief answer.

Let $\text{u}$ be a nonzero vector in $R^n$ and $W$ be the subset of $R^n$ defined by $W=\left\{{\text{x:u}}^T\text{x}=0\right\}$.

a) Prove that $W$ is a subspace of $R^n$.

b) Show that $\dim \left(W\right)=n-1$.

c) If $\theta =\text{w}+c\text{u}$, where $\text{w}$ is in $W$ and $c$ is a scalar show that $\text{w=}\theta$ and $c=0$. [Hint: Consider ${\text{u}}^T\left(\text{w}+c\text{u}\right)$.

d) If $\left\{{\text{w}}_{\text{1}}\mathrm{.....},{\text{w}}_{n-1},\text{u}\right\}$ is a basis for $W$, show that $\left\{{\text{w}}_{\text{1}}\mathrm{.....},{\text{w}}_{n-1},\text{u}\right\}$ is a basis for $R^n$[Hint: Suppose that $c_1{\text{w}}_{\text{1}}+\mathrm{...}+c_{n-1}{\text{w}}_{n-1}+c\text{u}=\theta$.