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
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.
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?
@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