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    Ivan Savov
    @ivan.savov_gitlab
    Test 1 2 3
    Ask your questions about math here $\sin\theta$
    rsparkman
    @rsparkman
    Hi Ivan, I had a question regarding material on page 26 of your Mathematics (green) book. On page 26 you're walking through how one can rewrite an equation by changing variables (in the example, because someone has a square root phobia - i can relate :))
    Specifically related to the last paragraph of that section which starts: "We now see that the solutions..." Is the expectation that the reader would understand what is to follow from reading the text, or is it more of an example to illustrate the point you're making and the reader will understand later when you go through that material?
    3 replies
    luismorana
    @luismorana
    Hello Ivan. First post from me and thanks for giving some room to people that are learning Math. I have a questions regarding the application of the inverse function to the example in the bottom part of page 26 and beginning of page 27 of the PDF version 5.4 or your Math and Physics book. You explain how to find x by applying the inverse function. You apply the inverse function to both sides and then simplify. How you do this simplification?...."since 5^x cancels log x...?." Would you kindly explain this process.....Thanks, Luis
    2 replies
    Ivan Savov
    @ivan.savov_gitlab
    narrative.png
    They key to understanding what is going on here is to consider the table of inverses given on the previous page:
    table-of-inverses.png

    Each row in that table shows a function and its inverse function. If f(x) "does" something to a number, then the inverse function f^{-1} is the "undo" operation.

    The example when f(x)=2x and f^{-1}(x) = x/2 is the simplest... the function f multiplies numbers by two, so the inverse operation is to divide by 2.

    If you apply f followed by the inverse you should get back the number you initially started from:
    f^{-1}( f(x) ) = x

    Ivan Savov
    @ivan.savov_gitlab
    The example contains the function log_5(x) (logarim base 5 of x) whose inverse function is the function 5^x (five to the power x, a.k.a the exponential function base 5), so the procedure I followed is to apply the inverse 5^x to undo the log_5(x)
    2 replies
    Ivan Savov
    @ivan.savov_gitlab

    Essentially we're using the f^{-1}( f(q) ) = q rule which holds for any quantity q, and in this case f(x) = log_5(x) and f^{-1}(x) = 5^x and q = 3+sqrt(6*sqrt(x)-7)

    @luismorana Lemme know if this helps or I can clarify further.

    luismorana
    @luismorana
    Thanks Ivan. I really appreciate your explanation. Yes it did clarified the issues I had and was able to understand the main idea. I will move on with the book and I am sure when I hit the chapter on logarithms, the complete idea will be consolidated. Thanks again for your time and consideration. BTW, how I am supposed to know when the new version of the book is available? Luis.
    2 replies
    Supman911
    @Supman911

    I have a pretty lengthy question, consisting of two parts.

    Lets say we are given a linear transformation T:V→W where V=R^n and W=R^m.

    I have learned that R^n=R(M)⊕N(M), where R(M) is the row space of the matrix M and N(M) is the null space of the matrix M, and matrix M is the matrix associated with the linear transformation T.

    I understand how the input space R^n contains the null space, as we input some vectors and they are mapped to the zero vector. But I do not understand how the input space contains R(M). What part does R(M) play in the input of T?

    Another question. Is the output space R^m made up of only C(M) or C(M) and N(M^T), where N(M^T) is the left null space of M. I am asking this because I have learned that for a matrix ∈Rm×n, the right space R^n=R(M)⊕N(M) and the left space R^m=C(M)⊕N(M^T). So is it the same with linear transformations? If R^m=C(M)⊕N(M^T), then what part does N(M^T) play in the output space or output of T?

    Just ended chapter 6 and had these confusions. Thank you.

    6 replies
    Supman911
    @Supman911
    Another question: I managed to prove that different eigenvalues give us linearly independent eigenvectors. But the converse is not true, yes? We can have linearly independent eigenvectors with the same eigenvalue. For example, the identity matrix I_2 has two linearly independent eigenvectors with the same eigenvalue 1.
    2 replies
    Ivan Savov
    @ivan.savov_gitlab
    Hi @Supman911, both great questions! Also good job finding all the Unicode characters for the math concepts.. I'm impressed. I've replied to your questions in threads.
    Supman911
    @Supman911
    Thanks! Sorry for disturbing you constantly. What about N(M^T)? How does N(M^T) play a part in our output space R^m? For any matrix-vector product Mv = w, we have that w is in the span of C(M) and so it makes sense that C(M) plays a part in the output space. But how does N(M^T) play a part in the output space?
    3 replies
    Supman911
    @Supman911

    Sir Ivan, thank you for responding but I am still having some issues.

    First of all, I can see that the row space gets mapped to the column space, the nullspace to 0. But why isn't any vector getting mapped to null space of A^T?

    Second of all, totally different question. Each vector space must contain the zero vector and since the 4 fundamental subspaces are vector spaces, they all contain the 0 vector, yes?

    3 replies
    Supman911
    @Supman911
    @ivan.savov_gitlab
    1 reply
    Ivan Savov
    @ivan.savov_gitlab

    @Supman911 (and anyone else reading the LA book).

    Today I found a really interesting visualization of the algebra operations like vector-vector product, matrix-vector, and matrix-matrix products.

    Check it out here: https://github.com/kenjihiranabe/The-Art-of-Linear-Algebra/blob/main/The-Art-of-Linear-Algebra.pdf Highly recommended, especially if you have a color printer!

    This is inspired by Gilbert Strang's new LA "short course" which you can watch on youtube here https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE21EJCxwmWvvek

    Supman911
    @Supman911
    Thank you!