- #1

karush

Gold Member

MHB

- 3,269

- 5

On the set of vectors

$\begin{bmatrix}

x_1 \\ y_1

\end{bmatrix}\in \Bbb{R}^2 $

with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by

$$\begin{bmatrix}

x_1 \\ y_1

\end{bmatrix} \oplus

\begin{bmatrix}

x_2 \\ y_2

\end{bmatrix}

=

\begin{bmatrix}

x_1 + x_2 \\ y_1y_2

\end{bmatrix}$$

and a scalar multiplication by

$$ k \odot

\begin{bmatrix}

x \\ y

\end{bmatrix} =

\begin{bmatrix}

k x \\ y^{k}

\end{bmatrix}.

$$

Determine if this is a vector space.

If it is, make sure to explicitly state what the $0$ vector is.

OK the only the only thing I could come up with was $2+2=4$ and $2\cdot 2=4$

and zero vectors are orthogonal with $k=2$

$\begin{bmatrix}

x_1 \\ y_1

\end{bmatrix}\in \Bbb{R}^2 $

with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by

$$\begin{bmatrix}

x_1 \\ y_1

\end{bmatrix} \oplus

\begin{bmatrix}

x_2 \\ y_2

\end{bmatrix}

=

\begin{bmatrix}

x_1 + x_2 \\ y_1y_2

\end{bmatrix}$$

and a scalar multiplication by

$$ k \odot

\begin{bmatrix}

x \\ y

\end{bmatrix} =

\begin{bmatrix}

k x \\ y^{k}

\end{bmatrix}.

$$

Determine if this is a vector space.

If it is, make sure to explicitly state what the $0$ vector is.

OK the only the only thing I could come up with was $2+2=4$ and $2\cdot 2=4$

and zero vectors are orthogonal with $k=2$

Last edited: