why is linear algebra linear?

What is linear and linearity?

 In this post, I will talk about what is linear algebra. Before we start, Let's make sure what linear is. Linear is very simple. If you can explain some function in a straight line, that is linear. linear is a straight line. If you see any curve or discreteness on the line, that is not linear. Linearity is also easy to be explained. You can think of it as proportionality like the relationship of acceleration and velocity, and the relationship of the weight of a thing and gravitation to earth. 

 Linearity has two properties. f(x) is a linear function. 
  1. Additivity : f(x) + f(y) = f(x+y)
  2. homogeneity : f(ax) = af(x) , when the degree of homogeneity is 1
 To talk about why linear algebra is linear, we need to know what linear equation is. in the Wikipedia, linear algebra is explained as the branch of mathematics concerning linear equation such as a1x1 + ... + anxn = b. The linear equation is also easy to understand. As the word itself, It is the equation that consists of linear.
  1. ax = b
  2. ax+by = c
  3. a1x1+a2x2+ ... + anxn = y
The first equation is obviously the linear equation. the value b will be increased by the argument x proportionally. The second one looks a little bit tricky but not really. The value c will also change according to the argument x and y proportionally. So the second one is also a linear equation. Like the sense, we can understand the last equation is the linear equation. 

why linear algebra is linear.

 Ok. It's time to talk about why linear algebra is linear. If you have searched on Wikipedia about linear algebra, you'd have seen the description like "Linear algebra is the branch of mathematics concerning linear equation." And we already know what linear equation is. So I hope that you don't feel difficult that I bring the third equation above in order to explain it. Let' say you are shopping in a grocery store. And you pick two unlabeled apples, one unlabeled banana up and those are 3$. And then you pick one the same apple and two the same bananas again up and pay 3$. After that, if someone comes to you and asks the prices of each fruit you bought, can you give the right answer that how much it is? you may have a simultaneous equation with two linear equations in order to give the right answer.
  • 2*apple + 1*banana = 3$
  • 1*apple + 2*banana = 3$
To solve the problem is so easy. Because you picked only two kinds of fruits. But what if you pick like 30 kinds of things? Then how many equations do you need to solve? It would be a lot. If there is a way that you can change quit complicated simultaneous equations into a pretty simple linear equation look like Ax=y, why not try to use that? The way is linear algebra. Linear algebra allows us to make Ax=y  out of a1x1+a2x2+ ... + anxn = y. More details, 



 As you have seen, we can convert any a lot of combinations of linear equations into a linear algebra equation. And linear algebra equation looks like the shape of the linear equation. Ax=y will be your boy. This is why linear algebra is linear.


References

Comments

Post a Comment

Popular Posts