We can solve a system with two variables like this:

$3x + 2y = 12 \\ x + 5y = 20$

with __replacing__ or __substituting__ – no problem. But as soon as you have more than three variables this manual, error prone approach will put you in a world of pain.

That's where Linear Algebra saves the day – among other things, it gives us tools to batch mathematical operations and represent them as a single transform.

`Notation`

`Cheatsheet with Definitions`

Matrices are caps $A$ , scalars $a$ , vectors $\vec{a}$

## Movie Analogy

`Vectors`

, lines in n-dimensional space, are actors in the movie.`Matrices`

squish, move, flip, rotate or collapse vector spaces. They move the plot forward`{Determinants, Eigenthings, Dot Products, Spans, …}`

talk about relationships of vectors among themselves and to matrices – they write the reviews. For example, a determinant can tell you how much a transform squishes or distorts a vector space. Eigenvectors tell you if a transform changes a vector's spans (*usually true for rotations or shears*) and much more

## Overview

`3Blue1Brown's Essence of Linear Algebra`

is an absolute **must see**`Interesting Take From the Trenches`

** Some Explorables**:

`Coursera Interactive`

– great interactive on transforms, but you will need a Coursera **Login**. Gives a good feeling for

*dilations, rotations, shears, reflections, projections*.

`Determinant, Eigenvector, Span`

`Matrix Multiplication`

Term | Describe | Formalize |
---|---|---|

Vector | Newsflash**: A vector is sectrely a transform | $\vec{x} |

Matrix | A transform in the making. A matrix is actually not something that is somewhere in vector space. It's a function, and isn't instantiated. Intuition: Vector $x$ lands on $v$ after we apply transform $A$ (*also*: $T_A$) | $ A\vec{x} = \vec{v} $ |

Inverse | $String + Z $ Reverts a transform | $I: I(A) = A \\ A^{-1}A = I $ |

Unit Vector | Unit vectors describe the axis of a coordinate system. Denoted $\hat{i}, \hat{j}, \hat{k} $…. for x, y, z ... axis. A good way to imagine what a Matrix $A$ does is to see each column $C$ as where the respective unit vector ($C_1 \to \hat{i}$) ends up after the transform. | |

Identity | Unit vectors lead to the *Identity* transform (*and vice versa*). If $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ = $\hat{i}$ and $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ = $\hat{j}$ are the unit vectors then $I$ is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ | $I \\ I \ of \ A^{2x2} = [\hat{i} \ \hat{j}]$ |

Tensor | Generalizes $Matrix \to Tensor_{rank2}$, $Vector \to Tensor_{rank1}$ | |

Dot Product | Tells you how two vectors directions relate – opposed, co-linear, orthogonal (right angle) | $\vec{a} * \vec{b} = x |

Determinant | Describes if a transforms preservers (*positive*), reverses (*negative*), or collapses (*0*) the orientation of the n-space. | $|A| \ or \ det(A) \\ det(\begin{bmatrix} a & c \\ b & d \end{bmatrix})= ad -bc $ |

Dot Product | Tells you how two vectors directions relate – opposed, co-linear, orthogonal (right angle) | $\vec{a} * \vec{b} = x$ |

Rank | Although the determinant tells us if a matrix collapses n-space, it couldn't tell us how many dimensions it collapsed. Full-rank means, the matrix doesn't change dimensions $D$. Rank1 means the output is $D = 1$ and so on. | |

Span of Vector | Line that passes through a *vectors* origin and tip. Or: Infinite extension of the vector.Most transforms will knock most vectors of their span, but the vectors that stay on span are called ***eigenvectors***. | |

Eigen | It's said that the most frequent use for Eigen-x is when we want to reduce complexity. This compression goes by many terms depending on field like PCA, SVD... | $Av = \lambda v \\ Av – \lambda Iv = 0 \\ (A – \lambda I)v = 0 \\ det(A – \lambda I) = 0 $ |

**Transform in Code**

Look at the Transforms ${A_1, A_2, A_3}$ . What happens to the unit vectors? What is the determinant? What would inverse the transform? Try to find the solutions and play with the code.

### Python Play

`TODO`