Question

7. Show that V = R2 with the given operations and o is not a vector space. 11 [][i:1-1 3]..[]-[] Y1

Answer 1

Consider $V=\mathbb{R}^2$ With the operations $\oplus\; and \; \odot$ defined as.

$\begin{bmatrix} x_1\\ y_1 \end{bmatrix} \oplus \begin{bmatrix} x_2\\ y_2 \end{bmatrix}=\begin{bmatrix} x_1-x_2\\ y_1 -y_2 \end{bmatrix}, c\; \odot \begin{bmatrix} x_1\\ y_1 \end{bmatrix}=\begin{bmatrix} cx_1\\ cy_1 \end{bmatrix}$

Consider two vectors

$u=\begin{bmatrix} 0\\ 0 \end{bmatrix} \; and \; v= \begin{bmatrix} 1\\ 1 \end{bmatrix} \; in \; \mathbb{R}^2$

Then we have

0 1 u U= = 0-1 0-1

and

$v\oplus u=\begin{bmatrix} 1\\ 1 \end{bmatrix}\oplus \begin{bmatrix} 0\\ 0 \end{bmatrix}=\begin{bmatrix} 1-0\\ 1-0 \end{bmatrix}=\begin{bmatrix} 1\\ 1 \end{bmatrix}$

It is observed that

$u\oplus v\ne v\oplus u$

Therefore  $\oplus$ is not commutative.

Hence is not a vector space.