2cm In the higher dimensions you cannot see everything, so you must have something, some tool, to guess or formulate things. And the tool was algebra, unquestionably algebra.
– Heisuke Hironaka (1931 - )
Fields medallist
In linear algebra, the main objects of study are linear transformations between vector spaces. The first conceptual breakthrough you are expected to make is that given a linear transformation, the matrix associated to it depends on the choice of basis of the vector space. Next, we study two natural subspaces which are associated to any linear transformation: the kernel and the image. These subspaces, and their dimensions, contain information about the behaviour of the linear transformation which does not depend on the choice of basis.
In applications, there is often a “better” basis than the standard one. For example, sometimes a non-standard basis is computationally more efficient, or perhaps it makes it easier for humans to interpret data. We will learn how to convert a matrix (or coordinates of a vector) from one basis to another.