First-principles math. External links live in resources.md. Linear-algebra ideas also support ../ai/notes.md and ../numpy/notes.md.
- Studies non-empty sets of mathematical objects together with algebraic operations
- A measurable property of an object or system
- The size or extent of a quantity
- Quantity that has magnitude only (no direction)
- A specific location (point)
- Straight path bounded by two distinct endpoints
- Contains all points between them
- A line extends infinitely in both directions — a segment does not
- Line segment from the center of a circle or sphere to its perimeter
- Quantity with both magnitude and direction
- As a list of numbers:
$[1,\ 3,\ 5,\ 10,\ 20]$ - In physics diagrams: an arrow whose length is magnitude
- Closed 2D shape with three sides, three angles, three vertices
- Interior angles sum to
$180\degree$
- Always has one
$90\degree$ angle - Relative to angle
$\theta$ :-
Opposite — side opposite
$\theta$ -
Adjacent — side next to
$\theta$ (not the hypotenuse) - Hypotenuse — side opposite the right angle (longest)
-
Opposite — side opposite
- Ratios of sides of a right triangle
- For a fixed
$\theta$ , each ratio is independent of triangle size
| Ratio | Definition |
|---|---|
| Sine | |
| Cosine | |
| Tangent |
- Sine and cosine values lie in
$[-1,\ 1]$
-
$a^{2} + b^{2} = c^{2}$ (right triangle;$c$ is the hypotenuse)
- Product of two vectors that yields a scalar:
$\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|,|\mathbf{b}|,\cos\theta$ -
$\theta$ — angle between the vectors
- Stub — magnitude $|\mathbf{a}|,|\mathbf{b}|,\sin\theta$ and direction (right-hand rule) to be filled.
- Set of all points in a plane at a fixed distance from a center
- Radius — center to edge
- Diameter — through the center, edge to edge
- Circumference — distance once around
$\dfrac{\text{circumference}}{\text{diameter}} = \pi \approx 3.14159\ldots$ $C = \pi d = 2\pi r$
$A = \pi r^{2}$
- A circle can be rearranged into a shape whose area is easier to see (classic pizza / triangle argument), yielding the same
$A = \pi r^{2}$
| Name | Meaning |
|---|---|
| Tangent | Line that just touches the circle at one point |
| Secant | Line that cuts the circle at two points |
| Chord | Segment joining two points on the circumference |
| Arc | Portion of the circumference |
-
Sector — “pizza” slice (region bounded by two radii and an arc)
-
Quadrant — sector with a
$90\degree$ central angle
-
Quadrant — sector with a
- Segment — region between a chord and its arc
- Semicircle — half a circle
- Circle not necessarily centered at the origin:
$(x - a)^{2} + (y - b)^{2} = r^{2}$ -
$(a,\ b)$ — center;$r$ — radius
- Rectangular array of numbers
-
Rows — horizontal:
$[1,\ 5,\ 67,\ 7,\ 8]$ - Columns — vertical entries
-
Rows — horizontal:
- Size written
$m \times n$ (rows × columns) — “down, then right” - Dimensions are set by row and column counts (e.g.
$2\times 3$ ,$3\times 5$ )
- Each entry is an element
- Indexed by row and column subscripts:
$M_{2,3}$ is row 2, column 3
- $1$s on the main diagonal, $0$s elsewhere
- Square (
$n \times n$ ); written$I$ - Acts like algebraic
$1$ :$A I = I A = A$
- Nonzeros only on the main diagonal
- Diagonal matrix whose diagonal entries are all equal
- Lower — zeros above the main diagonal
- Upper — zeros below the main diagonal
- Multiply every element by the same coefficient
- Binary operation: product of two matrices is another matrix
- Result has as many rows as the first factor and as many columns as the second
- Used in graphics for rotation, scaling, and translation
- Also the core of attention-style scores in ../ai/notes.md
- Scalars, vectors, and matrices are tensors of increasing rank
- Multiple matrices stacked form a 3D (rank-3) tensor
- Number of dimensions (axes)
- Adding another stacked “layer” of matrices increases rank
- Stub — axioms and metric structure to be filled.