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…
1. THE CHANGE OF COORDINATE SYSTEM In linear algebra, we often get comfortable working with the “standard” basis—the familiar $x$ and $y$ axes. But what happens when the problem you’re trying to solve looks messy in one coordinate system but…
I often find that the most beautiful concepts are the ones that build bridges. In linear algebra, vector spaces can be quite abstract—lists of numbers or functions floating in a void. But once you equip that space with an Inner…
1. The Definition of Norm (Length) In standard Euclidean geometry (like high school math), we find the length of a vector using the Pythagorean theorem. In abstract linear algebra, we generalize this concept using the Norm. The Definition: Let \(V\)…
In Linear Algebra, choosing the right basis can be the difference between a five-minute calculation and a five-page headache. While any linearly independent set can act as a basis, not all bases are created equal. In this post, we explore…