# Vectors and Representation - Outline Todo: really make it clear the big idea you wish to capture! What is the mic drop? ### Think about… - What’s it for?: what’s the goal of your post - To allow someone new to the field to simply “load” into their mind the notion of vectors that I have spent so much time learning about myself. - Who’s it for?: who’s your ideal reader - Previous Nate - Who are you writing for? What are their concerns and goals? - They want to understand how new, innovative ideas are created. - What action do you expect your reader to take? What is the purpose of your content? - They should walk away with a better understanding of applied mathematics. They should be able to understand new algorithms more quickly, as well as come up with their own. * Why should they believe you? What makes your content credible? A track record? Research? * Clear, coherent writing that is factually correct Also look at [The Pyramid Principle](The%20Pyramid%20Principle.md) ### Writing Approach 1. Giving the summarizing idea first, but make sure you leave the reader wanting more! 2. Structure 1. Situation 2. Complication 3. Question 4. Answer ### Outline (V1) * DS Mental Model Principle * *Description* & *Representation* & *Spaces* * SCQ * Example problem, you are trying to describe something * A * Generality * LEAVING OFF: * Explain Vector Space * [The Spirit of Linear Algebra](The%20Spirit%20of%20Linear%20Algebra.md) - [Polynomials are vectors](Polynomials%20are%20vectors.md) - [Elements of Rn as Vectors](Elements%20of%20Rn%20as%20Vectors.md) - Jkun → text - Jkun → Any useful notes from SVD article * Drawbacks of generality * Combination & Decomposition * Linear combinations, Component spaces * Basis * Vectors are invariant under change of basis * Description is different than the thing itself ### Big Ideas to Capture - **The generality of linear algebra** - Simplicity of a vector space - only requires the **ability** to add and multiply elements by a scalar - [The Spirit of Linear Algebra](The%20Spirit%20of%20Linear%20Algebra.md) - [Polynomials are vectors](Polynomials%20are%20vectors.md) - [Elements of Rn as Vectors](Elements%20of%20Rn%20as%20Vectors.md) - **The "drawbacks of the generality"** - We often forget what we are dealing with * Treating objects on their own terms * Motivation: Why is this necessary? * Because it makes it *far* easier to understand why geometric notions come into play so frequently. * Also, in applied math your vectors are frequently *things*! They *mean something!* * [Singular Value Decomposition Part 1: Perspectives on Linear Algebra – Math ∩ Programming](https://jeremykun.com/2016/04/18/singular-value-decomposition-part-1-perspectives-on-linear-algebra/) * TODO: parse key ideas from notes on Jkun articles in notability * really highlight how frequently this occurs in ML * To understand many ML techniques, you must understand the underlying building blocks, why it works * [Linear Algebra 2j: Elements of ℝⁿ as Vectors - So Boring, yet so Important! - YouTube](https://youtu.be/yhN_t4Y7-UA?list=LL) * [Linear Algebra 2p: A Passionate Appeal - Treat All Objects on Their Own Terms! - YouTube](https://www.youtube.com/watch?v=kz-RoBiqxRw&list=LL&index=8&t=58s) - Avoiding conflating objects - Difference between geometric objects, sets, etc - Component spaces! - key role in allowing us to use Jkun approach - **Why does the generality of linear algebra occur?** - How is Rn isomorphic with all other spaces - [Site Unreachable](https://yutsumura.com/every-n-dimensional-vector-space-is-isomorphic-to-the-vector-space-rn/) - See Linear Algebra good notes ("N dimensional vector spaces are isomorphic to Rn") - Notation tends to make this worse (we frequently don't include basis vectors in our notation - see goodnotes Linear Algebra pg 4) - Pg 5 Linear Algebra Good Notes * **Decomposition** * Highlight that this is in large part what makes projection so easy! * Pg 5 Linear Algebra Good Notes * **Data Vectors and Matrices** * Only include parts of this that specifically touch on vectors, not transformations between then *yet*. That will be in next post. * Pg 16 Linear Algebra Good Notes * jkun svd post 1 * **Basis** * vectors can be expressed in terms of other vectors. A nice set of these from which all others in a space can be expressed from is a *basis* * Part of what makes a basis so useful is that it allows us to have *unique representations* of vectors in our space (see linear algebra text by kun, pg 145) * Chapter 14 of Hello Again, Linear Algebra (pg 138) → We 1) pick a basis and translate our real life problem into that basis 2) solve matrix problem 3) Interpret that solution in real life setting * change of basis * noteability ways to view a matrix * This touches nicely on *methods of description* * Good time to mention [Relationship between Translating Space and Translating Objects](Relationship%20between%20Translating%20Space%20and%20Translating%20Objects.md) * [Tensors for Beginners 1: Forward and Backward Transformations (contains error; read description!) - YouTube](https://www.youtube.com/watch?v=sdCmW5N1LW4&list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG&index=4&t=366s) * Notability section is very good - the key is that a matrix can be thought of as transforming a vector, or equivalently, changing the way we *describe* that vector to reflect different basis * The key thing to explain is how do you know the difference? Or are they the same? Can I just * [Change of basis in Linear Algebra - Eli Bendersky's website](https://eli.thegreenplace.net/2015/change-of-basis-in-linear-algebra/) any square invertible matrix can be seen as a change of basis matrix * **Axioms of vector space** * likely should start pretty early in post since so much is built on top of it * **Linear combinations** * Math the beautiful videos * notability: linear algebra, lines, geometry * Idea: in this post you should start to make clear the idea of building objects (vectors) out of other objects. Linear combinations are perfect for this. As is the idea of **span**. But only tease the notion of linear transformations. Don't feel the need to include them just yet! * **Jkun Linear algebra chapter 10** * need to review it all and add the key ideas * Idea about vectors: * Vectors are the things we measure, and the things doing the measuring (the rulers!) they work so well as rulers due to the way vector addition and linear combination and dimesion are defined * [Tensors for Beginners 2: Vector definition - YouTube](https://www.youtube.com/watch?v=uPbBDToXjBw&list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG&index=5&t=20s) * Vectors are *invariant* under change of coordinates * the description of the thing is different than the thing itself! * Movie recommendation. We have users (vector), movies (vector), but are the recommendations themselves vectors? ### Not sure where to put this yet [Linear Algebra Intuitions & Big Ideas](Linear%20Algebra%20Intuitions%20&%20Big%20Ideas.md) * Jkun math texte book * Linear algebra in notability and goodnotes * math ml pdfs * Tia Bradley * References to shape * Where should Linear combinations go? * Likely *here* but I am not 100% sure yet --- Date: 20220609 Links to: Tags: #review References: * []()