Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra.
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The perspective you show to any mathematical phenomenon is just mind-blowing and so interesting. I never thought matrices could mean so much more than a bunch of numbers representing an equation. Please write a book on Mathematics
[[3,0],[0,-2]] This matrix is adding a to y component to the unit vector [1,0] but how is it giving it a direction that our original never had? Also the matrix [[3,0],[0,-2]] transforms [0,0] to [0,0] and [0,1] or any [0,a] to [0,0] therefore points in this new space are not unique. How is this consistent with the stretching of the axis metaphor in the video?
woooow i learned matrices at the school and at the university they were very komplex und important and i had no idea what the fuck are they. All I knew that they are some numbers in brackets!
Thank you very much.
This guy gives great intuition but I wish he connected the math to the intuition more. After I get the intuition from him I always have to look at the math and see how his intuition falls from the math. It's an extra step for me that he could explain in his videos.
For instance, why does being closed under addition and multiplication mean to that the grid lines must be straight and parallel?
It’s sad how many math students will never be able to appreciate the beauty of linear algebra beyond just lists of numbers in square brackets. Videos like this need to be taught more often. Anyway, great job a usual!
I passed this course with out knowing the beautiful intuition behind it. To put it bluntly, I passed it with out knowing shit, just memorizing what and when I have to compute in order to get the right answer. Now I have built a basic intuition, I find this subject fascinating.
Great teacher. Thanks!!!
if i understand this right than you could you also divide the matrix to find out where the untransformed point would have been ? and if so what happens if you pick a matrix that transforms 2d space into a line and than pick a point not on that line to divide by the matrix? dose it involve the 3rd dimension in that case or is it impossible ? does it have any practical implications?
It seems that n by n matrices describe linear transformations in n dimensional vector space. However, if matrices are fundamentally based on describing linear transformations, than what linear transforms do non square matrices represent and how?
I think perhaps non square n by m matrices describe transforming a m dimensional vector into a n dimensional vector. If this is the case, are all the basis vectors of the input space linearly independent to any basis vectors of the output space? Is it impossible to visualize transforms described by non-square matrices as points sliding between points?
Now I understand why multiplication with identity matrix give us the same result, this is because identity matrix has the coordinations of unit vectors i and j, which by definition is the same vector as it is. I rearly comment on youtube videos but sir, you have my full love and respect for sharing your wisdom with us. Thanks for your efforts.
Sorry if im making a stupid question, but if matrices are like functions, which take vector inputs and outputs, can you build the graph of that function, and can you use calculus to find the derivative?
It is a transformation of space, but a kinda weird one. It transforms 3-dimensional space into 2-dimensional space. It takes the first basis vector for 3-space and sends it to the first column vector of the matrix, etc.
I felt like I had been trapped in a dark cave of eternal ignorance prior to discovering this gem of a lecture. Thank you so much for shedding new light on this abysmally dreaded topic. I'm looking forward to quality videos on tensors and tensor calculus :)
It's not very clear for me why the transformed vector can be expressed as that sum of the transformed basis vectors times the original vector's coordinates. In the video it started with "it turns out that..." but I didn't get the explanation for why. Could someone explain?
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