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.

**Answer:**

\(V\) 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

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.

**Answer:**

\(V\) 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

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.

**Answer:**

\(V\) 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

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.

**Answer:**

- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- 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

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- 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

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

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

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- 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

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:

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

**Answer:**

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

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.

**Answer:**

- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- additive inverses do not always exist

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.

**Answer:**

- vector addition is not associative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- 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

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- additive inverses do not always exist

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- scalar multiplication is not associative
- 1 is not a scalar multiplication identity
- scalar multiplication does not distribute over vector addition

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:

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

**Answer:**

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

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.

**Answer:**

- scalar multiplication does not distribute over scalar addition

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- additive inverses do not always exist

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.

**Answer:**

- 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

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.

**Answer:**

- additive inverses do not always exist

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:

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

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- additive inverses do not always exist

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.

**Answer:**

- additive inverses do not always exist

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.

**Answer:**

- scalar multiplication does not distribute over vector addition
- scalar multiplication does not distribute over scalar addition

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.

**Answer:**

- vector addition is not associative
- vector addition is not commutative
- scalar multiplication does not distribute over scalar addition