Lang, Strang, whats next... Aang? Welcome future airbenders! relevant resources: [[Matrix Alphabet]] ## Linear Algebra ### Vectors and Vector Spaces Linear algebra is all about [[Vector|vectors]], commonly represented as [[tuple|tuples]] and what happens when you [[vector addition|add]] and [[scalar multiplication|scale]] them. Vectors are [[Element of a Set|elements]] of [[vector space|vector spaces]], and as such, can be combined to form more vectors within the same spaces. A [[subset]] of a vector space that also meets the requirements of being a vector space is called a [[subspace]] A sum of scaled vectors are a [[linear combination]], and [[the set of all linear combinations of a set of vectors forms a subspace]] Vector addition is pretty straight forward, but their multiplication(s) are a little less obvious. The [[dot product]] results in a scalar, and it has a few uses. 'Squaring' a vector by taking the dot product of a vector and itself, results in the square of the [[vector length]], so a simple square root afterwards will give the magnitude. [[non-degeneracy property of dot product]] [[unit vector]] [[perpendicular vectors have a zero dot product]] [[dot product of unit vectors is cosine of angle between]] [[cosine of angle between vectors]] [[Cauchy-Schwarz-Buniakowsky Inequality]] [[Triangle Inequality]] [[function space]] [[continuous functions are a subspace of function space]] [[differentiable functions are a subspace of continuous functions]] [[intersections of subspaces are also subspaces]] [[linearly dependent vectors]] [[linearly independent vectors]] [[standard basis vectors]] [[basis for vector space]] [[uniqueness of combination of independent vectors]] [[coordinate with respect to basis]] [[maximal subset of independent vectors]] [[maximal subsets form a basis]] [[larger sets of vectors than the maximal are dependent]] [[bases for a vector space have equivalent cardinality]] [[dimension of a vector space]] [[subspace of equal dimension is equal]] [[subspace never has greater dimension]] [[completing a basis]] [[sum of subspaces]] [[direct sum of subspaces]] [[existence of direct sum]] [[dimension of direct sum]] [[direct product of vector spaces]] ### Matrices [[matrix]] [[matrix addition]] [[scalar matrix multiplication]] [[matrix multiplication]] [[matrix multiplication distributes over matrix addition]] [[scalars commute and associates through matrix multiplication]] [[matrix multiplication is associative]] [[matrix transpose]] [[square matrix]] [[diagonal matrix]] [[identity matrix]] [[system of linear equations]] [[homogeneous system]] It's easy to burn out on linear algebra... Hoping to return soon. [[problems catalog]]