Thanks Kumar, glad it clicked.
G(x , y , z) = F(x + y + z) = F(x) + F(y) + F(z)
Very good (as always), however I think you do not explain the crucial aspect of dimension independence of a vector space. The “mini arithmetic” addition above can never actually happen as each element represents an independent dimension. This is one of the most difficult concepts to understand IMO, especially when you think of polynomial or function vector spaces.
Thanks Dimitris, great feedback. Down the road I’d like to do a follow-up on linear algebra, with independent vectors as the focus. I think the idea of a spreadsheet gets the notational/mechanical elements out of the way, so we can then begin exploring the underlying concepts (just what is an input, anyway?). Appreciate the thoughts!
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Hi Kalid, I couldn’t quite figure out why F(x) = x+3 is not linear. After all, y=x+3 is a straight line meeting the y axis at (0,3), and with a slope of 1. This definition of a straight line (i.e. linear) is different from the definition of ‘linear’ you gave. Am I missing something here. BTW, I am not a math major 
but your explanations to complex math are quite intuitive. thanks
Hi Kumar,
Great question. The term “linear function” actually refers to two separate (but related) concepts (see http://en.wikipedia.org/wiki/Linear_function for more detail).
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A polynomial of degree 0 or 1, i.e. f(x) = ax + b
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a “linear map”, meaning a function that has the properties that scaling the inputs scales the outputs, f(ca) = cf(a), and adding the inputs adds the outputs, f(a + b) = f(a) + f(b).
The function f(x) = x + 3 meets the first definition (polynomial of degree 1), and it is a straight line when drawn. But it doesn’t have the linear input/output relationship. For example, f(1) = 4, but f(2) = 5. We doubled the input, but did not double the input.
The main reason a line is not “linear” (in the linear map sense) is because of that + b term, which is +3 in our case. That +3 is the same amount, no matter how the input changes.
The two meanings are easily-confused, and did confuse me for a long time! Linear algebra refers to deal with behavior of functions that are linear maps.
Thanks. By the way, thanks to your post, I finally understood the reason behind the movie name “Matrix”. It is about, matrices, the “transformations” of “real space” into “virtual space”. It just dawned on me a moment ago when watching the matrix reloaded! Granted it is a science fiction movie, but still, for the movie producers or whoever, to have come up with that very apt name is really amazing (because of the required mathematical insight).
Love your flair Kalid!
This post had me laughing hysterically! Especially the part about “The survivors are…”
Widh I’d seen something like this when I was first exposed to matrices, maybe I wouldn’t have run the other way. Imagine my dismay when, in pursuit of my true love of physics, I encountered the Riemanian metric tensor and had to go back and learn all that matrix stuff I’d ignored for years. Oh yeah, and then there’s moment of inertia. For so long I had been taught it is a single number, but no it just had to be a matrix. It made sense though. Matrix transforms one vector to another. Torque = I * alpha, moment of inertia transforms angular acceleration (vector) to torque (vector). Now if those pesky eigenthingies would just leave me alone!
[…] 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.