B Buffs

In linear algebra, we often want to find the simplest set of building blocks that can construct every single element in a space without any redundancy. This set is called a Basis. We are going to define a Basis using…

We have already explored the building blocks of Linear Algebra: first, Linear Combinations (understanding individual vectors), and second, Linear Independence (understanding sets of vectors). Now, we are ready to dive into the “verbs” of Linear Algebra: Linear Transformations and The…

Linear Transformation is essentially a specific rule for moving vectors from one vector space to another vector space while keeping the underlying structure intact. In this post, we will break down the formal definition, look at real-world examples (including Linear…

If you are diving into linear algebra, you have likely noticed a shift. We start with relatively comfortable grids of numbers (matrix), but suddenly, we find ourselves in the world of “abstract vector spaces” (\(V\) and \(W\)). It can feel…

Unlike simple addition, where we just add corresponding numbers, matrix multiplication involves a complex process of multiplying rows by columns and summing them up. Why make it so complicated? The answer lies in Linear Transformations. Matrix multiplication is not just…

Introduction In basic algebra, the concept of an “inverse” is intuitive. If you have a function that doubles a number, \(f(x) = 2x\), the inverse function is the one that “undoes” that action, bringing you back to where you started.…

Part 1: The Tools – Elementary Row Operations In our previous post, we defined what it means for a matrix to be “invertible.” Now, we need to learn how to actually find that inverse. We can’t just divide by a…