V1 - Vector spaces


Example 1 πŸ”—

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

Example 2 πŸ”—

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

Example 3 πŸ”—

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

Example 4 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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

Example 8 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 13 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 19 πŸ”—

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.

Answer:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

Answer:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 28 πŸ”—

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

Example 29 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 33 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 36 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

Answer:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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.

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 46 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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 πŸ”—

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:

\(V\) 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