Chapter Text
Miruka and Tetra decided to meet up in a coffee shop to do some math. Neither of them had plans for what math they wanted to do; they figured that they would simply explore and let the math guide them wherever it may.
“Oh, hey! Tetra, Miruka! Nice to see you,” Yuri said, having noticed them enter the coffee shop.
“Yuri, it’s nice to see you,” Tetra said.
“Aren’t you a little young to be drinking coffee?” Miruka asked, adjusting her glasses.
“I’m here for the smoothies and pastries, actually,” Yuri replied. “Whatcha up to? Wait, lemme guess… Math?”
“Got it in one!” Tetra said.
“What are you going to study?” Yuri said.
“We haven’t decided yet. If you have the time, you could join us and I could show you something, if you want,” Miruka said.
“Sure, that would be great!” Yuri said. “Can you teach me about matrices? I heard my cousin talking about them, but I don’t see what the big deal is. It’s a rectangle of numbers, so what?”
“Matrix theory, also called linear algebra, has a lot of depth and beauty to it. I can show you, once I grab some coffee.” Miruka said.
“I’m pretty interested in this, too!” Tetra said. “I’ve multiplied matrices and things like that, but I haven’t seen anything beautiful or elegant about them yet. Just seems like a pain, really.”
After the three girls got their orders, they sat down at a table and Miruka pulled out her notebook and pencil.
“Let’s start with the basics. You said that matrices were rectangles of numbers, right Yuri?”
“Yes, is that wrong?” Yuri asked.
“Maybe not wrong, but a bit oversimplified,” Miruka replied. “Matrices are an array of symbols or expressions. They can be numbers, but they don’t have to be. For example, this matrix
Is as valid as this matrix.”
“Make sense?” Miruka asked.
“Wait, what are some of those symbols in the first one?” Yuri asked.
“The exact meaning of the symbols isn’t very important in this case. The point is that mathematical variables and expressions can be used in a matrix, not just numbers. By the way, the things inside the matrix are called ‘entries’ or ‘elements’,” Miruka explained.
“You’ll learn about more of these symbols once you get to high school,” Tetra assured Yuri.
Tetra paused for a moment. “I forgot, how do I refer to a specific element in a matrix?”
“You refer to each element as ai,j, where i is the row the element is in and j is its column. Well, I say that you use ai,j, but you can use other letters too, like bi,j to refer to another matrix.” Miruka said.
“Oh, I see, it would be confusing if you said ai,j was an element in matrix A then turned around and said that it was an element of matrix B.” Tetra said.
“Exactly,” Miruka said with an approving nod.
“Okay, so we’ve defined what a matrix is. But what do you do with them?” Yuri asked.
“Before we can do matrix operations, we need to discuss one more thing: the matrix’s dimensions.” Miruka said. “The matrices I showed you before are both 2x3 matrices because they have two rows and three columns. But matrices can have other dimensions, of course. To generalize, an mxn matrix has m rows and n columns. Yes, Tetra, you can use letters other than m and n. They’re just some of the more commonly used ones for matrix dimensions.”
“So now we can do matrix operations?” Tetra said.
“Yes. I’ll start easy with addition,” Miruka said. “To add two matrices, they must have the same dimensions. Then you just add corresponding elements. So, you add these two matrices like this.”
“You can also do what’s called ‘scalar multiplication’,” Miruka said. “A scalar is a value that isn’t a matrix, so it’s a number, variable, or expression. When you do scalar multiplication, you just multiply every entry in the matrix by the scalar, like this.”
“Can you subtract matrices?” Tetra asked.
“Yes, I’ve already defined it,” Miruka said.
“But you didn’t mention.... Oh! You can multiply one of the matrices by -1 and then add, and that’s equivalent to subtraction!” Tetra said, a little too loudly.
“Exactly,” Miruka said with a smile.
“Wow, you’re really on top of this, Tetra,” Yuri said. “I wouldn’t have noticed that so quickly.”
“Don’t be hard on yourself, it takes practice to be able to notice things like that. I just refer back to the definitions, and think about what’s possible,” Tetra said.