Assignments 1

1. Let
   $V$ be a vector space over a field
   $F$ with vector addition
   $\oplus$ and scalar multiplication
   $\odot$. Assume
   $\mathbf{u}\mathbf{,}\mathbf{v}\mathbf{,}\mathbf{w}\in V$, solve the following system of equations: (You must indicate at each step which properties are applied.)
   
   $$\begin{array}{rl}& \mathbf{u}\oplus \mathbf{v}\oplus (2\odot \mathbf{w})=\mathbf{0},\\ & (2\odot \mathbf{u})\oplus (2\odot \mathbf{v})\oplus \mathbf{w}=\mathbf{0},\\ & (3\odot \mathbf{u})\oplus (4\odot \mathbf{v})\oplus (2\odot \mathbf{w})=\mathbf{0}.\end{array}$$
2. Let
   $V$ be the vector space defined as
   
   $$V=\{(x_1,x_2,x_3)|x_1,x_2,x_3\in {\mathbb{R}}^{+},x_1+x_2+x_3=1\},F=\mathbb{R},$$
   with
   $\oplus$ and
   $\odot$ defined as
   
   $$\begin{array}{rl}(x_1,x_2,x_3)\oplus (y_1,y_2,y_3)& =\frac{1}{x_1{y}_1+x_2{y}_2+x_3{y}_3}(x_1{y}_1,x_2{y}_2,x_3{y}_3),\\ \alpha \odot (x_1,x_2,x_3)& =\frac{1}{x_1^{\alpha }+x_2^{\alpha }+x_3^{\alpha }}(x_1^{\alpha },x_2^{\alpha },x_3^{\alpha }).\end{array}$$
   Let
   $\mathbf{u}=(\frac{1}{2},\frac{1}{4},\frac{1}{4})$,
   $\mathbf{v}=(\frac{1}{4},\frac{1}{2},\frac{1}{4})$ and
   $\mathbf{w}=(\frac{1}{4},\frac{1}{4},\frac{1}{2})$, find
   $\alpha$ and
   $\beta \in \mathbb{R}$ such
   $\mathbf{w}=(\alpha \odot \mathbf{u})\oplus (\beta \odot \mathbf{v})$.
3. Let
   $V={\mathbb{R}}^2$ with
   $\oplus$ and
   $\odot$ defined as
   
   $$(a_1,a_2)\oplus (b_1,b_2)=(a_1+b_1,a_2{b}_2),\alpha \odot (a_1,b_1)=(\alpha a_1,b_1).$$
   Show that
   $V$ is not a vector space.
4. Exercise 2.6 at p.11.
5. Exercise 3.3 at p.17.

TA solution

• Assignment1_solution