What is Linear algebra: Definition and 999 Discussions
Linear algebra is the branch of mathematics concerning linear equations such as:
a
1
x
1
+
⋯
+
a
n
x
n
=
b
,
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:
(
x
1
,
…
,
x
n
)
↦
a
1
x
1
+
⋯
+
a
n
x
n
,
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
In fact, it WAS a homework couple of years ago, and I've solved it, kind of (below). I still would like to find a cleaner solution.
Here is what I did.
Let's say, the apples are labeled, and their weights are ##x_1, x_2, ...##. He takes out the apple #1 and finds that, e.g., ##x_2+x_5+x_9+... =...
Hello,
I need some advice because I just can't figure out how to solve the problem. I could try to make the determinant triangular by adding all the b together, but that doen't seem a good way of solving the problem. Is there any direction I should be thinking of?
Thanks
This is problem 28 from chapter 3F "Duality" of Axler's Linear Algebra Done Right, third edition.
I spent quite a long time on this problem, like a few hours. Since there is no available solution, I am wondering if my solution is correct.
One assumption in this problem is that...
My question is about item (b).
(b)
Here is what I drew up to try to visualize the result to be proved
The general idea, I think, is that
1) ##(\text{null}\ T)^0## and ##\text{range}\ T'## are both subspaces of ##V'=L(V,\mathbb{F})##.
2) We can show that they have the same dimension.
3)We...
I will use a proof by cases.
Case 1: dim V = dim W
Then ##T=T|_V## is an isomorphism of ##V## onto ##W##. The reason for this is that it is possible to prove that if ##T## is surjective, which it is, then it is also injective and so it is invertible (hence an isomorphism).
Case 2: dim V < dim...
I was stuck when I started writing this question. I think I solved the problem in the course of writing this post.
My solution is as follows:
Consider any basis ##B## of ##V## that includes ##v##: ##(v, v_2, ..., v_n)##.
##L(V,W)##, where ##\dim{(V)}=n## and ##\dim{(W)}=m## is isomorphic with...
The Math challenge threads have returned!
Rules:
1. You may use google to look for anything except the actual problems themselves (or very close relatives).
2. Do not cite theorems that trivialize the problem you're solving.
3. Do not solve problems that are way below your level. Some problems...
The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help.
I’ll put the screen shot in the comment part
Let $$ X \in R^{m*n} $$ where m=n with rank(X)<m then there is at-least one equation which can be written as a linear combination of other equations. Let $$ \beta \in R^{n} $$.
$$ X\beta=y $$
Suppose we have x<m independent equations (the equations are consistent) formed by taking the dot...
Here is an example of the decomposition for a 2 x 2 matrix
We have ##2^2=4## determinants, each with only #n=2# non-automatically-zero entries. By "non-automatically-zero" I just mean that they aren't zero by default. Of course, any of ##a,b,c##, or ##d## can be zero, but that depends on the...
TL;DR Summary: we are given a set of coefficient matrices (shown below) and we need to determine whether they are in REF, RREF, or neither.
Hello! I am having a lot of trouble identifying these matrices, and using the criteria checklist is not helping very much. Here is what I am working with...
\begin{pmatrix}
2 & 4 & 6 \\
3 & 5 & 8 \\
1 & 2 & 3
\end{pmatrix}
Using the row operations, R2<-- R2-3R1 R3<-- R3-R1 we find the row echelon form of the matrix.
\begin{pmatrix}
1 & 2 & 3 \\
0 & -1 & -1 \\
0 & 0 & 0
\end{pmatrix}
Based on the definition of row space in the book Í am...
It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they...
In classical mechanics, it seems like solving force equations are a question of finding a solvable system of equations that accounts for all existing forces and masses in question. Therefore, I'm curious if this can be mixed with reinforcement learning to create a game and reward function...
...Out of interest am trying to go through the attached notes,
My interest is on the highlighted, i know that in
##\mathbb{z}/\mathbb{6z}## under multiplication we shall have:
##1*1=1##
##5*5=1## am assuming that how they have the ##(\mathbb{z}/\mathbb{6z})^{*}={1,5}## is that correct...
I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way?
In the...
My question is motivated by the proof of TH 5.13 on p 84 in the 2nd edition of Linear Algebra Done Right. (This proof differs from that in the 4th ed - online at: https://linear.axler.net/index.html chapter 5 )
In the proof we arrive at the following situation:
##T## is a linear operator on a...
I am intending to join an undergrad course in physics(actually it is an integrated masters course equivalent to bs+ms) in 1-1.5 months. The thing is, in order to take a dive into more advanced stuff during my course, I am currently studying some of the stuff that will be taught in the first...
Welcome to the reinstatement of the monthly math challenge threads!
Rules:
1. You may use google to look for anything except the actual problems themselves (or very close relatives).
2. Do not cite theorems that trivialize the problem you're solving.
3. Have fun!
1. (solved by...
For exercise 3 (2),
,
The solution for finding the eigenvector is,
However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here?
Many Thanks!
Thread moved from technical math section, so there is no homework template.
(∀λ∃ℝ)
-x + y - z = 1
-2x + 10y + (2λ + 6) = 6
3x + 11y + (λ2+6)z = 5λ - 1
after gaussian elimination I have this:
-1 4 -2 | 1
0 1 λ | 2
0 0 λ(λ-1) | 5λ
So, for λ=0 ⇒ ∞ solutions, for λ=1...
Assume a table A(3x3) with the following:
A [ 1 2 1 ]^T = 6 [ 1 2 1 ]^T
A [ 1 -1 1 ]^T = 3 [ 1 -1 1 ]^T
A [ 2 -1 0]^T = 3 [ 1 -1 1]^T
Find the Eigenvalues and eigenvectors:
I have in mind to start with the Av=λv or det(A-λI)v=0....
Also, the first 2 equations seems to have the form Av=λv...
Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect.
So, I can define...
Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
The usual theorem is talking about the linear operator being restricted to an invariant subspace:
I had no problem understanding its proof, it appears here for example: https://math.stackexchange.com/questions/3386595/restriction-operator-t-w-is-diagonalizable-if-t-is-diagonalizable
However...
The classical "power method" for solving one special eigenvalue of an operator works, in a finite-dimensional vector space, as follows: suppose an operator ##\hat{A}## can be written as an ##n\times n## matrix, and its unknown eigenvectors are (in Dirac bra-ket notation) ##\left|\psi_1...
Hi,
If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A##
Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##.
Is there way to show...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything.
At about 5:50, he states that "The array for Q is...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material.
Basically, is everything below this correct?
In summary of the derivation of the...
I refer to the video of this page, where there is a description of Galilean relativity that is meant to be an introduction to SR, making the comprehension of the latter easier as a smooth evolution from the former.
All the series is in my opinion excellent, but I think that this aspect is...
Top-Down-Perspective:
At first I am quite sure that the problem is not solvable since there are that many unknowns.
But my Approach would be to create a linear function with P1 and P2 and then set it equal to the function that gets me the impact location of P3 and then solve it by b3.
Thanks...
I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary:
Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...
We have three Random variable or vector A,B,C. Condition is A & B are independent as well as B & C are independent RVs . But A & C are the same random variable with same distribution . So How can determine E{ABC}. Can I write this E{ABC}= E{AE{B}C}?
I came across the following problem somewhere on the web. The original site is long gone.
The problem has me stumped. May be sopmeone can provide some insight.
(The problem seems too simple to post in the "Linear/Abstract Algebra" forum.)
The Cost of Beer
It was nearing Easter, and a group...
I learned that for a bilinear form/square form the following theorem holds:
matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form.
Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
I'm trying to review some high school maths and work my way to Calculus and Linear Algebra, and I found these three translations of Japanese maths textbooks translated by the AMS and edited by Kunihiko Kodaira. The AMS links to them are:
https://bookstore.ams.org/cdn-1669378252560/mawrld-8/...
Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and
\begin{align}
\left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt.
$$
0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.