Linear Algebra Explained: Vectors, Matrices, and the Magic Between

For a long time, I studied only for marks, memorizing everything just to pass exams. I understood the material temporarily, but it was only for the test — not because I was passionate about it.

This approach continued until grade 12, where I realized that studying just for marks was pointless. It became clear that true learning wasn’t about grades, but about understanding and personal growth.

Determined to change, I decided to stop studying for marks and focus on truly understanding the material. That’s when I went back to the basics and began learning from scratch.

The Essence of Linear Algebra: Breaking it Down

When I first encountered linear algebra, I thought it was all about x’s and y’s — just points on graphs, static and simple. But that’s just the surface.

Linear algebra, in reality, begins with numbers. The 1’s, 2’s, 3’s — the numbers we all know. These numbers can form lists, either in rows or columns. When you arrange numbers in rows or columns, they form vectors — one-dimensional lists of numbers.

When we combine these vectors, they form matrices. A matrix is essentially a collection of vectors. These matrices can be n-dimensional. While you might start thinking in terms of 2D (two-dimensional) or 3D (three-dimensional), matrices can exist in any number of dimensions — even higher, like 5D or more! We can't easily visualize these higher dimensions, but they’re just as real and useful.

Now, let’s talk about operations like matrix addition, subtraction, and multiplication. Matrices represent transformations. Imagine you have a plane with horizontal and vertical lines. You can manipulate these planes in different ways, using matrix operations to transform them. By transforming, I mean changing magnitude and direction.

This is the core idea behind linear algebra: it’s all about manipulating magnitudes and directions, not about shapes.

If you’ve studied physics, you’ve probably heard about vectors — they have magnitude (how long or powerful something is) and direction (where they point). That’s why vectors are used in physics: they describe forces, velocities, and other quantities that have both magnitude and direction.

In the context of linear algebra, this idea of magnitude and direction is extended. Matrices and vectors help us represent transformations in multi-dimensional spaces. We're no longer just flipping shapes around; we’re understanding how to change their size, stretch or shrink them, or rotate them, in more than just 2D or 3D space.

Rather than focusing solely on numerical computations — which can now be done in seconds using a computer or even a mobile app — what truly matters for a student of mathematics, engineering, or any related field is visualization. Visualization is the real key to understanding. In Linear Algebra especially, geometric intuition plays a vital role. It helps you see what’s happening — not just calculate. Whether you're dealing with vectors, transformations, or matrices, being able to visualize directions, magnitudes, and spaces brings everything to life. And this isn’t limited to Linear Algebra — visual thinking is fundamental across all areas of mathematics and physics. Cultivate it, and you’ll never look at equations the same way again.

I hope you have better understanding of linear algebra now to deeply dive into it!!