An isomorphism is a very strong relationship between structures and in essence states that the two structures are identical in their behavior under operations, that sets of images have the same characteristics, etc. You acknowledge that φ_γ mapping is an isomorphism, so it maps bases of R(T)...
OK, without column spaces - Obviously [A] is the matrix representation of L_A in bases F^m and F^n. If c is any vector in V and [c] is its representation in F^n, then [Tc] = [T][c] = [A][c] = (L_A)[c] . So you have Tc is in range of T if and only if (L_A)[c] is in range of L_A. So those two...
By definition of [T], the column space of [T] = A is the range of T. Similarly, by definition of L_A, it's range is the column space of A. So range of T is the same space as range of L_A and they must have same dimension. That's it.
I've scanned a page out of my textbook and highlighted the portion of the proof I don't quite follow. I've been staring at this on and off for a day and for some reason it just doesn't click why the argument shows that R is uniquely determined by W. I see the author is proving an implication and...
I'm working out a problem that requires me to proof a result by induction. I have worked out what I think is a correct proof, but I would like for somebody to look over it and give me feedback.
Part 1: Give a reasonable definition of the symbol \sum_{k = m}^{m + n}{a}_{k}
I first define...
I'm in my 5th semester undergrad for bachelor of engineering in mechanical engineering, but I realized that I like mathematics much more than I enjoy my ME or any other applied courses that I have to take. I'm thinking about applying for graduate study in mathematics at top US schools (Berkeley...
You need to know how high the workers are to figure out how much work is done in lifting the rope itself too (all 30 ft initially), which gets shorter as the workers pull on it. Since work is a scalar, split it into work done on pulling the weight (no integration necessary) and work done in...
I think I have worked out a correct proof. I would appreciate it if anyone could review this and criticize if necessary.
I have already established that S is nonempty and bounded above so I will not dwell on this fact anymore. As in my first post, by the law of trichotomy, the least upper...
Thanks for the reply, but I was looking for something even more elementary - no limits and no squeeze theorem. Is this proof possible to do with even more basic concepts? I'm curious because Apostol is very thorough and I'm confident he wouldn't write this if it couldn't be done with the...
I'm doing self study out of Apostol's Calculus vol. I and I got stuck trying to prove what the author writes is easy to verify, but I can't get my head around it. Basically, this is the problem statement from page 31, last paragraph:
Given a positive real number x, let a[SIZE="1"]0 denote the...