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

edited February 2020

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

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1.
edited February 2020

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|\$ 
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2.
edited February 2020

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. 
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3.
edited February 2020

$$\mathbb{R ^ 3} = \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$ 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.
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4.
edited February 2020

$$\mathbb{R ^ n} = \mathbb{R} \times \mathbb{R} \times \ldots \mathbb{R}$$ 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.
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5.
edited February 2020

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.

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 counting numbers = n-dimension complex Cartesian space \$$\mathbb{R}^n\$$, \$$\mathbb{Q}^n\$$ and \$$\mathbb{C}^n\$$ are examples of vector spaces. 
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6.
edited February 2020

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. 
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7.
edited February 2020

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. 
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8.

Now let's do some math with vectors.

Comment Source:Now let's do some math with vectors.
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9.
edited February 2020

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

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} \$ 
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10.
edited February 2020

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} \$
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11.
edited February 2020

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} \$
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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} \$
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13.

What we have here is an algebra of vectors.

Comment Source:What we have here is an _algebra_ of vectors.
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14.
edited February 2020

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} \$ 
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15.
edited February 2020
Comment Source:Fin * * * [Next: Matrices, dot products, and matrix multiplication](https://forum.azimuthproject.org/discussion/2479/matrices-dot-products-and-matrix-multiplication)