## V1 - Vector Spaces (ver. 1)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 2)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (3(x_1+x_2), 2(y_1+y_2))\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 3)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (5cx,5cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 4)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2+2x_1x_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 5)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 6)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+6, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 7)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+5, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 8)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+6, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 9)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2+2x_1x_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 10)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (3(x_1+x_2), 4(y_1+y_2))\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 11)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (3cx,2cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 12)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (4cx,3cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 13)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (4x_1x_2, y_1+2y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 14)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 15)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2+4x_1x_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 16)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (4cx,2cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 17)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (3cx,3cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 18)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (3(x_1+x_2), 4(y_1+y_2))\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 19)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 20)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (4cx,4cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 21)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (1x_1x_2, y_1+2y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 22)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (2x_1x_2, y_1+2y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 23)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 24)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (5cx,3cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 25)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (5cx,5cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 26)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (4(x_1+x_2), 2(y_1+y_2))\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 27)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (4x_1+x_2, y_1+3y_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 28)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (5cx,5cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 29)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 30)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 31)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 32)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (c^3x, c^3y).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 33)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 34)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2+4x_1x_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 35)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (4x_1x_2, y_1+3y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 36)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+3, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 37)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (cx,cy-4c+4).\\ \end{align*}

(a) Show that scalar multiplication is associative, that is:

$a\odot(b\odot (x,y))=(ab)\odot(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 38)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (1x_1x_2, y_1+2y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 39)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+3, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 40)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+2, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 41)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 42)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 43)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (3x_1x_2, y_1+3y_2)\\ c \odot (x,y) &= (cx,0).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- additive inverses do not exist

- 1 is not a scalar multiplicative identity element

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 44)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1x_2,y_1y_2)\\ c \odot (x,y) &= (x^c,y^c).\\ \end{align*}

(a) Show that there exists an additive identity element, that is:

$\text{there exists }(w,z)\in V\text{ such that }(x,y)\oplus(w,z)=(x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- additive inverse elements do not always exist

## V1 - Vector Spaces (ver. 45)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 46)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+4, y_1+y_2)\\ c \odot (x,y) &= (cx,y^c).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 47)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2+3, \sqrt{y_1^2+y_2^2})\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that vector addition is associative, that is:

$\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- no additive identity element exists

- scalar multiplication does not distribute over vector addition

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 48)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (3x_1+x_2, y_1+2y_2)\\ c \odot (x,y) &= (cx,cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- vector addition is not commutative

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition

## V1 - Vector Spaces (ver. 49)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (2cx,2cy).\\ \end{align*}

(a) Show that scalar multiplication distributes over scalar addition, that is:

$(c+d)\odot(x,y)=c\odot(x,y)\oplus d\odot (x,y).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

- scalar multiplication does not distribute over vector addition

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative

## V1 - Vector Spaces (ver. 50)

Let $$V$$ be the set of all pairs $$(x,y)$$ of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= (x_1+x_2, y_1+y_2)\\ c \odot (x,y) &= (c^2x, c^3y).\\ \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

$c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2).$

(b) Explain why $$V$$ nonetheless is not a vector space.

$$V$$ is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: