📄️ Vector Spaces and Bases
A Vector Space is a bit abstract. It is a set $V$ of all objects of interest that can be added together or scaled 'safely' -- this is the Linearity in Linear Algebra. For something to be called a Vector Space, it must satisfy some axioms.
📄️ Matrices
A Matrix encodes an operator's 'instructions' on how to transform vectors in a space based on the coordinate system (Basis Vectors) used. Different system $\implies$ a different matrix.