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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 = {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

## Comments

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:

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

`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|\\]`

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.`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.`

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

`\\(\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.`

\(\mathbb{R ^ n} = \mathbb{R} \times \mathbb{R} \times \ldots \mathbb{R}\) 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 avector 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.

`\\(\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.`

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

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

`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 counting numbers = n-dimension complex Cartesian space \\(\mathbb{R}^n\\), \\(\mathbb{Q}^n\\) and \\(\mathbb{C}^n\\) are examples of vector spaces.`

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.

`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.`

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.

`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.`

Now let's do some math with vectors.

`Now let's do some math with vectors.`

Vector addition:

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

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

Vector subtraction:

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

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

Scaling a vector by a coefficent:

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

`Scaling a vector by a coefficent: \\[ 10 * \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} = \begin{bmatrix} 100 \\\\ 200 \\\\ 300 \\\\ \end{bmatrix} \\]`

Computing the negative of a vector:

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

`Computing the negative of a vector: \\[ -1 * \begin{bmatrix} 10 \\\\ 20 \\\\ 30 \\\\ \end{bmatrix} = \begin{bmatrix} -10 \\\\ -20 \\\\ -30 \\\\ \end{bmatrix} \\]`

What we have here is an

algebraof vectors.`What we have here is an _algebra_ of vectors.`

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:

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

`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} \\]`

Fin

Next: Matrices, dot products, and matrix multiplication

`Fin * * * [Next: Matrices, dot products, and matrix multiplication](https://forum.azimuthproject.org/discussion/2479/matrices-dot-products-and-matrix-multiplication)`