# Elements of $\mathbb{R}^n$ as Vectors Consider the following objects: * Geometric Vectors (directed segments) * Polynomials (funny symbols on a piece of paper) * Elements of $\mathbb{R}^n$ (sets of numbers) One can only marvel at how different these types of objects are! What is amazing is that each of these objects has two key properties in common that allow them to all be analyzed via the tools of linear algebra: 1. They can be **added** together to produce another element of the same kind. 2. They can be **multiplied** by numbers to produce another element of the same kind. With that said, by studying $\mathbb{R}^n$ we will eventually see that we are studying all types of vector spaces (since all vector spaces can be mapped to $\mathbb{R}^n$). ### Break the association of elements of $\mathbb{R}^n$ with geometric vectors Can an element of $\mathbb{R}^n$ (say when $n = 3$) be "stuck" in a particular subset of $\mathbb{R}^n$? The answer if of course yes! Consider the following sets of vectors: $ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + \begin{bmatrix} -2 \\ 7 \\ -6 \end{bmatrix} = \begin{bmatrix} -1 \\ 9 \\ -3 \end{bmatrix} $ $ 7 \begin{bmatrix} 4 \\ 0 \\ 12 \end{bmatrix} = \begin{bmatrix} 28 \\ 0 \\ 84 \end{bmatrix} $ We see that these vectors have the **property** that if we *add them together* (the first two vectors of the first equation) or *multiply by a scalar* (the vector on the left hand side of the second equation) we end up with a vector that has the third element being $3$ times the first element! Put another way: > Our vectors $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} -2 \\ 7 \\ -6 \end{bmatrix}$ both have their third element being $3$ times the first. When we add them together that **property** is **preserved**. Likewise, $\begin{bmatrix} 4 \\ 0 \\ 12 \end{bmatrix}$ also has its third element being $3$ times the first. So, if we have elements that are contained in the subspace of $\mathbb{R}^3$ we cannot get out of this subspace. We are *stuck* within that set where the last entry is $3$ times the first entry. We call that set a [Subspace](Subspace.md). However, and this is very important, in a sense there is absolutely **nothing geometric** about this space! It simply is the space of all triplets of numbers. Now, there are isomorphisms/[homomorphisms](https://en.wikipedia.org/wiki/Homomorphism) that exist between geometric vectors and triplets of real numbers, but on their own triplets of numbers have no geometric properties whatsoever! See more on this [here](https://youtu.be/yhN_t4Y7-UA?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&t=383). ### Treat All Objects on Their Own Terms Geometric vectors are *not* pairs of numbers (i.e. $n$-tuples). However, because there exists an isomorphism/homomorphism between the two we often switch between them interchangeably. Why is this an issue? The challenge is that we lose sight of the fact that they are indeed different things, which means we lose intuition about the **generality** of linear algebra, as well as what we are trying to represent. Does this actually present a problem in the real world? I would argue *yes*. The reason is that when you are trying to come up with *new ideas* or *new approaches* that make use of linear algebra and the objects it deals with, if you don't have a good understanding of ***what the mathematics are describing*** then it becomes challenging to come up with sound new ideas. --- Date: 20220606 Links to: [Linear Algebra MOC](Linear%20Algebra%20MOC.md) Tags: #review References: * [Linear Algebra 2j: Elements of ℝⁿ as Vectors - So Boring, yet so Important! - YouTube](https://www.youtube.com/watch?v=yhN_t4Y7-UA&list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&index=15) * [Linear Algebra 2p: A Passionate Appeal - Treat All Objects on Their Own Terms! - YouTube](https://www.youtube.com/watch?v=kz-RoBiqxRw&list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&index=24) * [Linear Algebra 2i: Polynomials Are Vectors, Too! - YouTube](https://www.youtube.com/watch?v=xhTciBubSfM&list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv&index=12)