[…] 不同成分之间必须是可合成的。思暮雪必须能被分离并重制(饼干？没那么夸张。谁有想要一堆碎屑呢？).不同成分在分离和结合的时候必须是线性的。 […]
[…] 不同成分之间必须是可合成的。思暮雪必须能被分离并重制(饼干？没那么夸张。谁有想要一堆碎屑呢？).不同成分在分离和结合的时候必须是线性的。 […]
Hi Kalid,
I disagree with your analysis of the principle of homogeneity in your above example.
A function f(x) is homogeneous if f(nx) = nf(x). To use your example:
f(x) = x + 3
f(2x) = 2(x+3)
For x = 1, then:
f(1) = 4
f(2*1) = 8
This should be true for all x.
Hi B. Rich, when evaluating the function you need to replace “x” with the value. So,
f(2x) = (2x) + 3
f(1) = 1 + 3 = 4
f(2) = 2 + 3 = 5
Very beautiful ! Thank you …
Hi,
I have a quick question about
“The determinant is the “size” of the output transformation. If the input was a unit vector (representing area or volume of 1), the determinant is the size of the transformed area or volume.”
If I have
[A = \begin{pmatrix} 2 & 1 \
0 & 1 \end{pmatrix}$ ]
and then feed the column vector (1, 1) into this operation, I get (3, 2).
The determinant of A is |A| = 2. So in what sense is the area of (3, 2) twice the area of (1, 1)?
Corrigendum: I get (3, 1) and not (3, 2). The question remains though.
@Determinator, @Mauricio: Whoops, I wasn’t clear enough. Let me clarify. Imagine an x-y axis. A unit square would be determined by two vectors, one on the x-axis (1, 0) and one along the y-axis (0, 1). In a matrix this is {{1 0}, {0 1}} which indeed has unit area:
Take another matrix, such as {{2,1},{0,1}}. The determinant is 2. Before doing the math, I know my original unit area will be transformed to some set of vectors that sweep out an area of 2.
Wolfram alpha shows the result:
which has area 2:
We can see that one vector is unchanged, but the other has been skewed, increasing the area of the total to 2. If my original vector was something like {{5,0},{0,1}}, with area 5, I know the result would be 10 after being transformed:
Hope this helps.
I agree with the person linking linear algebra to far more advances spaces like Sobolev spaces, Hilbert spaces.
Problem as always in these tutotials is proofs.
Math is about proving theorems. So to in LA.
In Belgium, 1st year at University LA class, the same.
Examination tests is on proving theorems.
So. Proofs (building) books are essential to maths. If you can’t take that hurdle, forget it. Analysis, heavy proofs, algebra the same.
Try once to explain a proof theorem in LA.
It will help a lot of people. Math is not about calculating. Leave solving systems of linear equations to the computer.
yes, I understand. Thanks for the clarification.
Thank you so much for this - having read and used your imaginary numbers post in my recent introduction for year 12, I thought ‘I wonder if he’s done anything on matrices?’ Watched Derek Holt’s lectures on Linear Algebra over the summer (they’re very good), but for a really intuitive introduction I can’t ask for better than this page. Granted, it won’t be a full description, but what I really need is an intuitive hook to get started with, and this was definitely it! My year 12 Further Maths group thank you!
Awesome, really glad to hear it helped :). Linear algebra befuddled me for a while because I always associated with “advanced” operations, like rotating a robotic arm in 3d or solving a giant system of equations. No – we can just take an everyday example, like having a stock portfolio and updating it based on some event. Seeing it as a ‘mini spreadsheet’ helped me wrap my mind around the use cases, which can of course expand into to the fancy vector operation stuff.
@Determinator
I’m also confused about what he says with he determinants and the unit vector!
First of all a unit vector is not a vector that has an area of one: Unit vector - Wikipedia
And multiplying a matrix A by a unit vector does not result in a vector of area = det(A), as you can see by simple examples.
So, I’m still confused about what a determinant means.
But wikipedia helps a little:
Also, look at
The area of the paralelogram represented by the matrix is 2, which is the determinant of the matrix.
I like to think of Eigenvectors and eigenvalues from the perspective of a pitcher standing on the mound on a windy day. In the space between the pitcher and home plate there are many lines of force going in different directions created by the swirling winds. Some of these happen to line up perfectly with the direction the pitcher is throwing the ball. These are Eigenvectors. When the ball is thrown along this perfectly lined up vector it will gain speed in addition to the initial speed it was thrown with. This additional speed is the eigenvalue. The lines of force are contours in the space that influence the way the ball moves through that space.
@TI: Great analogy, thank you!
Can you come up with a real world problem using matrices, like you have a real world problem that actually happens out there in the world and then you take this problem you write it in terms of equations with x and y and z and then you make matrices and you solve the problem, most importantly find the inverse of that matrix and tell us what does that inverse matrix mean in terms of real world problems, what does it tell us about the real world problem we just broke down into equations, what does the inverse matrix mean in that sense.
I know how to find the inverse matrix and all that stuff but in the end I dont have any idea what the hell it means in the real world, it just looks like arbitrary made up nonsense number trick game.
@Fation
Economists use linear algebra and specifically matrix inversion all of the time. One classic example would be input - output analysis. The form would be something like Q = AQ + B. Q would be the overall quantity demanded of a good that is both an input in making other goods (The AQ portion) and is also sold to an end user on its own (the B portion). A is a matrix of “technical coefficients” that describes how much Q goes in to make the other goods. In linear algebra because of the nature of matrices you can’t simply divide one by the other. Inversion takes the place of division (it is even denoted A^-1 which is another way of saying it’s a divisor). Multiplying by an inverse is = to division. So in this example Q-AQ=B --> (I-A)Q = B --> Q = (I-A)^-1 * B. “I” being the identity matrix. By multiplying the inverse of (I-A) by B we get Q. So through inversion we discovered an independent expression for Q which is very useful for figuring out some stuff. This is just a simple example but it is used in practice for modeling in the real world. This link provides more detail if you’re interested http://www.math.unt.edu/~tushar/S10Linear2700%20%20Project_files/Davidson%20Paper.pdf
I am getting a headache on tensors. Please post a lesson about it.
Beautifully explained . Having always been baffled by matrices and determinants , I have to say that this is the best lesson that I have had
Wow. This post just blew my mind! I had 3 distinct ‘aha’ moments on this one post alone.
I’ve loved every single one of our posts, but this one is remarkably special because, I pretty much had other topics figured out. I knew how to fuction with imaginary numbers. I knew how calculus worked. Your posts gave me new ways of looking at those topics, new methadologies of grasping and picturising those topics, but without those new insights, I still had a functional understanding.
This topic—I had no clue what was going on. The way I was taught, there was no concept of why or how—we’re taught “This is just the way it is”. I found it very arbitrary, very “made-up”. Like we just defined the matrix multiplication to be that complex operation without any forethought. I knew there had to be reason, there had to be logic. For some reason, I could find none on the internet. And then I came across your post. And it friggin’ blew my mind. So thank you. I really owe you for such a strong foundation that I’ve created about all sorts of topics. And especially for this one. Don’t know what I would’ve done without you!