# Linear Algebra & Hyperplanes & Geometry  The big idea here is that we are trying to bridge between two views of a matrix: 1. As a linear transformation that maps from one vector space to another, as seen in this [3b1b clip](https://youtu.be/uQhTuRlWMxw?t=157). Here we see that $A$ transforms $\vec{x}$ to $\vec{v}$. 2. As a set of constraints (hyperplanes) whose packaging together (i.e. they are all occurring at the same time) means their intersection represents a point/vector, $\vec{x}$. Specifically, we have a hyperplane in the form of $2x + 5y + 3z = -3$. There is a large set of $x, y, z$ that fall on this hyperplane. We also have 2 others, with a large set of $x, y, z$ that fall on them. The set of $x,y,z$ that fall on all three hyperplanes (often just a single tuple) is our solution, and is represented as the vector $\vec{x}$. --- Date: 20220104 Links to: [Linear Algebra MOC](Linear%20Algebra%20MOC.md) Tags: References: * Linear algebra, lines, geometry - Notability note