Many of you have probably not just heard about but learned matrices in a university or a high school. When I was a freshman and took an algebra course at the university, I couldn’t understand the reason for learning matrices. The only questions I had in my mind were, “Why do people care about matrices? Why do we need to learn how to multiply them and find the inverse matrices?”, despite being a top student. Although, then I found out. So, let me show you how powerful matrices can be.
From the first look, a matrix is just a table of numbers. It doesn’t look impressive, and it’s still unclear why we need to represent numbers this way.
Maybe you’ve heard about the evident and straightforward application of matrices — they are used to solve linear systems of equations, which usually are of huge size (even order of millions). In brief, each system of linear equations can be represented as A x = b, where:
- A is the matrix of coefficients
- x is a column matrix of the unknowns
- b is the column matrix, which contains the constants of the right-hand sides of the equations
So, the solution (elements of the matrix x) can be found as x = A⁻¹ b. As you can see, here we need to know how to find the matrix product and the inverse matrix. However, it’s an obvious and basic example of matrices application, so we won’t talk much about it.
Actually, matrices are something more than just a table of numbers. It’s a “magic” tool that can help you to modify spaces.
Let’s consider the next triangle of three vectors in the Cartesian coordinate system:
What if we want to rotate this triangle by 90 degrees? The easiest way to achieve this is… to use matrices! First, let’s write down the coordinates of the corresponding vectors as a matrix, knowing that a = (0, 1), b = (2, -1), c = (-2, 0):
The rotation matrix by angle α looks the next way:
Hence, the 90° rotation matrix is
To rotate our triangle by 90°, we need to find the matrix product of R and A:
So, the transformed vectors are (-1, 0), (1, 2), and (0, -2):
So, this is a simple but impressive example of how matrices can help us rotate vectors in a specific basis of vectors. Actually, matrices can do much more than just rotation. For instance, let’s find the product of a random invertible matrix (let’s call it S) and our matrix A:
So, in the following image, you can see the stretched original triangle. The origin grid lines are dotted, and the grey lines represent the stretched grid lines.
As you can see, we can modify figures and surfaces using matrices. To apply multiple modifications to vectors, you should multiply them on the left-hand sight of the expression. Hence, if you want to stretch our triangle and then rotate it, the matrix of modified vectors will look the following way:
Here’s the result on the graph:
So, if you want to bring our new triangle back to its original state, you should apply the inverse matrices:
As you can see, matrices can describe intuitively understandable things in a precise mathematical way. Moreover, the set of all 2×2 invertible matrices forms a group under multiplication! If you don’t know what groups are, check out my article about them.
Indeed, the set of 2×2 invertible matrices (let’s call it M) satisfies all the properties of a group:
- For any two matrices in M, their product belongs to M. It’s obvious that the product of matrices is a matrix, but will it always be an invertible matrix? So, since det(A B) = det(A) det(B) and both det(A) and det(B) don’t equal 0, det(A B) is also not equal to 0 and A B is an invertible matrix
- For any three matrices in M, no matter in which order you apply the multiplication. So, (A B) C = A (B C)
Caution! For arbitrary matrices, AB ≠ BA. - The identity 2×2 matrix I is a neutral element in this set.
- For any invertible 2×2 matrix A, there’s always an inverse matrix A⁻¹ and A⁻¹ A = A A⁻¹ = I
Hence, the set of 2×2 invertible matrices is an excellent example of infinite, non-Abelian groups. And I hope you can now see how beautiful and powerful matrices can be. See you in my other posts!
References
[1] T. Panov (2019). Linear Algebra and Geometry
[2] A. Kostrikin, Y. Manin. Linear Algebra and Geometry
[3] Socratica. Abstract Algebra course https://www.socratica.com/subject/abstract-algebra
[4] 3Blue1Brown. Matrix multiplication as composition https://www.youtube.com/watch?v=XkY2DOUCWMU
