V1 - Vector Spaces


Example 1

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.

Answer.

\(V\) 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


Example 2

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.

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

- scalar multiplication does not distribute over scalar addition


Example 3

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 4

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.

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


Example 5

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.

Answer.

\(V\) 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


Example 6

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.

Answer.

\(V\) 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


Example 7

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.

Answer.

\(V\) 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


Example 8

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.

Answer.

\(V\) 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


Example 9

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.

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


Example 10

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.

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

- scalar multiplication does not distribute over scalar addition


Example 11

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 12

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 13

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.

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 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


Example 14

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.

Answer.

\(V\) 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


Example 15

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.

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


Example 16

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 17

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 18

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.

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

- scalar multiplication does not distribute over scalar addition


Example 19

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.

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


Example 20

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 21

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.

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 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


Example 22

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.

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 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


Example 23

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.

Answer.

\(V\) 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


Example 24

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 25

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 26

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.

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

- scalar multiplication does not distribute over scalar addition


Example 27

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.

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 commutative

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition


Example 28

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 29

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.

Answer.

\(V\) 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


Example 30

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.

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


Example 31

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.

Answer.

\(V\) 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


Example 32

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.

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


Example 33

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.

Answer.

\(V\) 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


Example 34

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.

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


Example 35

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.

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 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


Example 36

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.

Answer.

\(V\) 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


Example 37

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.

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


Example 38

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.

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 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


Example 39

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.

Answer.

\(V\) 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


Example 40

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.

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


Example 41

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.

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


Example 42

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.

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


Example 43

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.

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 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


Example 44

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.

Answer.

\(V\) 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


Example 45

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.

Answer.

\(V\) 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


Example 46

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.

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


Example 47

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.

Answer.

\(V\) 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


Example 48

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.

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 commutative

- vector addition is not associative

- scalar multiplication does not distribute over scalar addition


Example 49

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.

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

- 1 is not a scalar multiplicative identity element

- scalar multiplication is not associative


Example 50

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.

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