Options

[section1] Numbers and vectors - tanzer.trail1.post1

Next: Matrices, dot products, and matrix multiplication


There are various choices for what a 'number' means:

  • \(\mathbb{N}\) = set of all natural numbers = {0, 1, 2, ...}
  • \(\mathbb{N^+}\) = counting numbers = {0, 1, 2, ...}
  • \(\mathbb{Z}\) = integers = {..., -3, -2, -1, 0, 1, 2, 3, ...}.
  • \(\mathbb{Q}\) = rational numbers = all fractions p/q for integer p,q
  • \(\mathbb{R}\) = real numbers = all limits of infinite sequences of rationals
  • \(\mathbb{C}\) = complex numbers = all a + bi for real a,b
Tagged:

Comments

  • 1.
    edited February 14

    The next concept on our agenda is the Cartesian product of sets.

    For sets \(A\) and \(B\), their (Cartesian) product \(A \times B\) is the set of all possible pairs (a,b), for a in A and b in B.

    Example: suppose \(A = \lbrace 1, 2 \rbrace\) and \(B = \lbrace 100, 200, 300 \rbrace \).

    Then \(A \times B\) = \(\lbrace (1,100), (1,200), (1,300), (2,100), (2,200), (2,300) \rbrace\).

    This product has six elements.

    The size of the product of A and B is the product of their sizes:

    \[|A \times B| = |A| \times |B|\]

    Comment Source:The next concept on our agenda is the Cartesian product of sets. For sets \\(A\\) and \\(B\\), their (Cartesian) product \\(A \times B\\) is the set of all possible pairs (a,b), for a in A and b in B. Example: suppose \\(A = \lbrace 1, 2 \rbrace\\) and \\(B = \lbrace 100, 200, 300 \rbrace \\). Then \\(A \times B\\) = \\(\lbrace (1,100), (1,200), (1,300), (2,100), (2,200), (2,300) \rbrace\\). This product has six elements. The size of the product of A and B is the product of their sizes: \\[|A \times B| = |A| \times |B|\\]
  • 2.
    edited February 14

    What's \(\mathbb{R} \times \mathbb{R}\)? This is also known as \(\mathbb{R} ^ 2\).

    It's the set of all pairs of real numbers.

    It's the Cartesian plane, the set of all (x,y) pairs, for real x and y.

    Comment Source:What's \\(\mathbb{R} \times \mathbb{R}\\)? This is also known as \\(\mathbb{R} ^ 2\\). It's the set of all pairs of real numbers. It's the _Cartesian plane_, the set of all (x,y) pairs, for real x and y.
  • 3.
    edited February 14

    \(\mathbb{R ^ 3} = \mathbb{R} \times \mathbb{R} \times \mathbb{R}\) is the set of all (x,y,z) triples of real numbers. This is three-dimensional Cartesian space.

    Comment Source:\\(\mathbb{R ^ 3} = \mathbb{R} \times \mathbb{R} \times \mathbb{R}\\) is the set of all (x,y,z) triples of real numbers. This is three-dimensional Cartesian space.
  • 4.
    edited February 14

    \(\mathbb{R ^ n} = \mathbb{R} \times \mathbb{R} \times \ldots \mathbb{R}\) is the set of all n-tuples of real numbers. This is n-dimensional Cartesian space.

    The members of \(\mathbb{R^n}\) are called vectors, and \(\mathbb{R^n}\) is a vector space.

    To be more specific, we can refer to the members of \(R^n\) as n-vectors.

    An n-vector, then, is simply an n-tuple, a list, of n real numbers.

    Comment Source:\\(\mathbb{R ^ n} = \mathbb{R} \times \mathbb{R} \times \ldots \mathbb{R}\\) is the set of all n-tuples of real numbers. This is n-dimensional Cartesian space. The members of \\(\mathbb{R^n}\\) are called _vectors_, and \\(\mathbb{R^n}\\) is a _vector space_. To be more specific, we can refer to the members of \\(R^n\\) as n-vectors. An n-vector, then, is simply an n-tuple, a list, of n real numbers.
  • 5.
    edited February 14

    We could repeat the same construction, to get n-dimensional spaces using different kinds of numbers:

    • \(\mathbb{Q}^n\) = set of all n-tuples of rational numbers = n-dimensional rational Cartesian space
    • \(\mathbb{C}^n\) = set of all n-tuples of rational numbers = n-dimension complex Cartesian space

    \(\mathbb{R}^n\), \(\mathbb{Q}^n\) and \(\mathbb{C}^n\) are examples of vector spaces.

    Comment Source:We could repeat the same construction, to get n-dimensional spaces using different kinds of numbers: * \\(\mathbb{Q}^n\\) = set of all n-tuples of rational numbers = n-dimensional rational Cartesian space * \\(\mathbb{C}^n\\) = set of all n-tuples of rational numbers = n-dimension complex Cartesian space \\(\mathbb{R}^n\\), \\(\mathbb{Q}^n\\) and \\(\mathbb{C}^n\\) are examples of vector spaces.
  • 6.
    edited February 14

    However, \(\mathbb{N}^n\), which consists of all n-tuples of natural numbers, definitely exists, but it is not considered to be a vector space.

    That's because in a vector space, we expect certain operations to always be defined: we should be able to add two vectors to get another, take the negative of vector to get its 'opposite', and scale a vector by a number to get another vector.

    Let's choose a 3-tuple in \(\mathbb{N}^3\), say x = (4,2,3).

    If we try to take the negative of x, that would be the 3-tuple (-4,-2,-3). But that doesn't belong to \(\mathbb{N}^3\), where components must be non-negative.

    Comment Source:However, \\(\mathbb{N}^n\\), which consists of all n-tuples of natural numbers, definitely exists, but it is not considered to be a vector space. That's because in a vector space, we expect certain operations to always be defined: we should be able to add two vectors to get another, take the negative of vector to get its 'opposite', and scale a vector by a number to get another vector. Let's choose a 3-tuple in \\(\mathbb{N}^3\\), say x = (4,2,3). If we try to take the negative of x, that would be the 3-tuple (-4,-2,-3). But that doesn't belong to \\(\mathbb{N}^3\\), where components must be non-negative.
  • 7.
    edited February 14

    Now an n-vector may be displayed either as a row-vector, or a column-vector.

    As a row-vector:

    \[ \begin{bmatrix} 4 & 2 & 3 \end{bmatrix} \]

    As a column-vector:

    \[ \begin{bmatrix} 4 \\ 2 \\ 3 \\ \end{bmatrix} \]

    In general, we'll work with column vectors as the default representation.

    Comment Source:Now an n-vector may be displayed either as a row-vector, or a column-vector. As a row-vector: \\[ \begin{bmatrix} 4 & 2 & 3 \end{bmatrix} \\] As a column-vector: \\[ \begin{bmatrix} 4 \\\\ 2 \\\\ 3 \\\\ \end{bmatrix} \\] In general, we'll work with column vectors as the default representation.
  • 8.

    Now let's do some math with vectors.

    Comment Source:Now let's do some math with vectors.
  • 9.
    edited February 14

    Vector addition:

    \[ \begin{bmatrix} 10 \\ 20 \\ 30 \\ \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} = \begin{bmatrix} 11 \\ 22 \\ 33 \\ \end{bmatrix} \]

    Comment Source:Vector addition: \\[ \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} + \begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\\\ \end{bmatrix} = \begin{bmatrix} 11 \\\\ 22 \\\\ 33 \\\\ \end{bmatrix} \\]
  • 10.
    edited February 14

    Vector subtraction:

    \[ \begin{bmatrix} 10 \\ 20 \\ 30 \\ \end{bmatrix} - \begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} = \begin{bmatrix} 9 \\ 18 \\ 27 \\ \end{bmatrix} \]

    Comment Source:Vector subtraction: \\[ \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} - \begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\\\ \end{bmatrix} = \begin{bmatrix} 9 \\\\ 18 \\\\ 27 \\\\ \end{bmatrix} \\]
  • 11.
    edited February 14

    Scaling a vector by a coefficent:

    \[ 10 * \begin{bmatrix} 10 \\ 20 \\ 30 \\ \end{bmatrix} = \begin{bmatrix} 100 \\ 200 \\ 300 \\ \end{bmatrix} \]

    Comment Source:Scaling a vector by a coefficent: \\[ 10 * \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} = \begin{bmatrix} 100 \\\\ 200 \\\\ 300 \\\\ \end{bmatrix} \\]
  • 12.

    Computing the negative of a vector:

    \[ -1 * \begin{bmatrix} 10 \\ 20 \\ 30 \\ \end{bmatrix} = \begin{bmatrix} -10 \\ -20 \\ -30 \\ \end{bmatrix} \]

    Comment Source:Computing the negative of a vector: \\[ -1 * \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} = \begin{bmatrix} -10 \\\\ -20 \\\\ -30 \\\\ \end{bmatrix} \\]
  • 13.

    What we have here is an algebra of vectors.

    Comment Source:What we have here is an _algebra_ of vectors.
  • 14.
    edited February 16

    Here is an example of an equation in the algebra of vectors.

    Let \(x\) stand for an unknown vector in \(\mathbb{R^3}\).

    Now solve for \(x\) in the equation:

    \[ x + \begin{bmatrix} 10 \\ 20 \\ 30 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \]

    Comment Source:Here is an example of an equation in the algebra of vectors. Let \\(x\\) stand for an unknown vector in \\(\mathbb{R^3}\\). Now solve for \\(x\\) in the equation: \\[ x + \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} = \begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ \end{bmatrix} \\]
  • 15.
    edited February 17
    Comment Source:Fin * * * [Next: Matrices, dot products, and matrix multiplication](https://forum.azimuthproject.org/discussion/2479/matrices-dot-products-and-matrix-multiplication)
Sign In or Register to comment.