## V1 - Vector spaces

#### Example 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)&= \left(x_{1} + x_{2} + 3,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c^{3} x,\,c^{2} y\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c^{2} x,\,c^{2} y\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} + 2 \, x_{2},\,3 \, y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(4 \, x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2} + 3,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} x_{2},\,y_{1} + 4 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 2\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 6 \, c + 6\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 6 \, c + 6\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(4 \, x_{1} x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(4 \, x_{1} + x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(4 \, x_{1} + x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 3 \, c + 3\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 3 \, c + 3\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2} + 2,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2} + 5,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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:

• there is no additive identity element
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 3 \, c + 3\right) . \\ \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

#### Example 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)&= \left(3 \, x_{1} x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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:

• there is no additive identity element
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c^{3} x,\,c^{4} y\right) . \\ \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

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(3 \, x_{1} + 3 \, x_{2},\,6 \, y_{1} + 6 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(3 \, x_{1} x_{2},\,y_{1} + 4 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 4\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2} + 3,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(4 \, x_{1} + x_{2},\,y_{1} + 2 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(3 \, x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2} + 4,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 7 \, c + 7\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} + 2 \, x_{2},\,4 \, y_{1} + 4 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2} + 2,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(3 \, x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(4 \, c x,\,3 \, c y\right) . \\ \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 is not associative
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition

#### Example 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)&= \left(x_{1} + x_{2} + 4,\,\sqrt{y_{1}^{2} + y_{2}^{2}}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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:

• there is no additive identity element
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c^{3} x,\,c^{4} y\right) . \\ \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

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} + x_{2},\,y_{1} + 3 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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
• vector addition is not commutative
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(3 \, x_{1} x_{2},\,y_{1} + 2 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,0\right) . \\ \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 associative
• vector addition is not commutative
• additive inverses do not always exist
• 1 is not a scalar multiplication identity
• scalar multiplication does not distribute over vector addition
• scalar multiplication does not distribute over scalar addition

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(x_{1} + x_{2} + 1,\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,y^{c}\right) . \\ \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

#### Example 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)&= \left(2 \, x_{1} + 2 \, x_{2},\,4 \, y_{1} + 4 \, y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(x_{1} x_{2},\,y_{1} y_{2}\right) \\c \odot (x,y) &= \left(x^{c},\,y^{c}\right) . \\ \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 inverses do not always exist

#### Example 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)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y - 7 \, c + 7\right) . \\ \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

#### Example 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)&= \left(3 \, x_{1} + x_{2},\,y_{1} + y_{2}\right) \\c \odot (x,y) &= \left(c x,\,c y\right) . \\ \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: