Linear algebra Definition and 999 Threads

I've been reviewing some introductory tensor stuff, and I've come to the realization that some of the things tensors do confuse me. For example, the notes I'm reading say that the invariant interval is both ##S=\eta_{\mu\nu} x^\mu x^nu## and ##S=x^T \eta x##. Both of which are totally fine on...

We know that a Jordan canonical form is simply the matrix representation of an operator (whose characteristic polynomial splits) with respect to a special basis called a Jordan canonical basis. This basis consists of a disjoint union of cycles/chains of generalized eigenvectors. Take all the...

The way I've approached the problem so far is that I'm looking for the set ##\{x\in\mathbb R^3: ABx=x\}##, since the rotation acts as the identity on the axis of rotation. The set is the null space of ##AB-I##, and $$AB=\begin{pmatrix}\cos\psi&-\sin\psi&0\\...

Conditioning doesn't refer to hair conditioners in this case, but what happens to the solution of a system of linear equations ##Ax=b## when we change say the vector ##b## slightly to ##b'##. If the relative change in ##b##, that is ##\frac{\|b-b'\|}{\|b\|}##, is close to the relative change in...

Here ##A=B_1\oplus B_2\oplus \cdots\oplus B_k## means that ##A## is block diagonal with ##B_i## along the diagonal. A proof that I've seen goes as follows. The ##k=2## case boils down to realizing that if ##v\in\beta_1\subseteq\mathsf W_1##, then ##\mathsf T(v)## is a linear combination of...

Let ##y_1,y_2,\ldots,y_n## be linearly independent functions in ##C^\infty(\mathbb C)## and let ##y\in C^\infty(\mathbb C)##. Define the column vectors $$v(t)=(y(t),y'(t),\ldots,y^{(n)}(t))^t$$and $$v_i(t)=(y_i(t),y_i'(t),\ldots,y_i^{(n)}(t))^t\quad\text{for }1\leq i\leq n.$$ Consider the linear...

The theorem reads as follows: Proof. The proof is by induction on the number of columns of an ##m\times n## matrix ##A##. The base case is kind of clear (either the rref is the ##0## vector or ##e_1##). Suppose now that the rref of a matrix with ##k## columns is unique and consider ##A## with...

I assume the reader is familiar with row echelon form of a matrix and reduced row echelon form (rref) of a matrix, the latter which is unique for each matrix. For instance, the augmented matrix $$\begin{pmatrix} 1&0&1&1\\ 1&1&-1&2\\ 2&0&1&0 \\ 0&-1&1&-3\end{pmatrix}$$has rref...

First, I'm a beginner at this, so sorry if my code looks stupid. I'd be grateful for any feedback concerning this. I wonder if anything's redundant or can be improved? The idea of the "algorithm" is to start with the first column of the matrix, in particular the first entry below the diagonal...

I struggle with a certain computation. Consider a homogeneous linear differential equation with constant (complex) coefficients. Such an equation is associated with a polynomial ##p(t)##, which we can write $$p(t)=(t-c_1)^{n_1} (t-c_2)^{n_2}\cdots (t-c_k)^{n_k},$$where ##n_1,n_2,\ldots,n_k## are...

I'm not entirely sure how to approach part 1. I tried the following: Call the span of the two states S. We can decompose H = S + S_perp. We consider some projector P: H -> V where V is some subspace of H. I made a new operator P_s P P_s where P_s is the orthogonal projector onto S. However, I...

An incidence matrix reflects a relation on a set of ##n## objects ##1,2,\ldots,n## and is a square matrix with ##0##s on the diagonal, and ##0##s and ##1##s elsewhere. For example, consider the incidence matrix $$A=\begin{pmatrix}0&1&0&0\\ 1&0&0&1 \\ 0&1&0&1 \\ 1&1&0&0\end{pmatrix}.$$Since...

Consider the rank-nullity theorem. We want to prove that for a linear transformation ##\mathsf T:\mathsf V\to\mathsf W##, $$\operatorname{nullity}(\mathsf T)+\operatorname{rank}(\mathsf T)=\operatorname{dim}(\mathsf V).$$We have a basis ##\{v_1,\ldots,v_k\}## of the null space ##\mathsf...

Throughout, let ##\mathsf V## be a vector space (the concept of dimension has not been introduced yet). The statement that precedes the theorem below is that if no proper subset of ##T\subset \mathsf V## generates the span of ##T## (where, if I'm not mistaken, ##T## consists of two or more...

Given a vector ##\mathbf{r} = r^i e_i## where ##r^i## are the components, ##e_i## are the basis vectors, and ##i = 1, \ldots, n##. In matrix notation, \begin{equation*} \mathbf{r} = \begin{bmatrix} e_1 & e_2 & \ldots e_n \end{bmatrix} \begin{bmatrix} r^1 \\ r^2 \\ \vdots\\ r^n \end{bmatrix}...

For this problem, The solution for (a) is I am slightly confused for ##p \in W## since I get ##a_3 = 2a_1## and ##a_2 = 2a_0##. Since ##a_3 = 2b##, ##a_2 = 2a##, ##a_1 = b##, ##a_0 = a##. Anybody have this doubt too? Kind wishes

The eigenvectors of a hermitian matrix corresponding to unique eigenvalues are orthogonal. This is not too difficult of a statement to prove using mathematical induction. However, this case is seriously bothering me. Why is the dot product of the vectors not rightly zero? Is there something more...

Why are the eigenvectors of this hermitian matrix not checking out as orthogonal? The eigenvalues are certainly distinct. ChatGPT also is miscalculating repeatedly. I have checked my work many times and cannot find the error. Kindly assist.

How does one find the dual of a matrix? Thanks.

I'm reading Group Theory by A. Zee , specifically, chapter I.3 on rotations. He used the passive transformation in analyzing a point ##P## in space. There are two observers, one labeled with unprimed coordinates and the other with primed coordinates. From the figure below, he deduced the...

Hi, a doubt about the definition of vector space. Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled. Now the point is: if one...

I was stuck on this problem so I looked for a solution online. I was able to reproduce the following proof after looking at the proof on the internet. By this I mean, when I wrote it below I understood every step. However, it is not a very insightful proof. At this point I did not really...

I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?

Hello, I am currently self studying Linear Algebra using MIT lectures and the textbook Introduction to Linear Algebra by professor Gilbert Strang. I'm at the 16th lecture on Projection Matrices and Least squares approximation. The lectures are very informative, but I struggle a lot with...

Suppose ##9## is an eigenvalue of ##T^2##. Then ##T^2v=9v## for certain vectors in ##V##, namely the eigenvectors of eigenvalue ##9##. Then ##(T^2-9I)v=0## ##(T+3I)(T-3I)v=0## There seem to be different ways to go about continuing the reasoning here. My question will be about the...

Here is one proof $$\forall u\in U\implies Tu\in U\subset V\implies T^2u\in U\implies \forall m\in\mathbb{N}, T^m\in U\tag{1}$$ Is the statement above actually a proof that ##\forall m\in\mathbb{N}, T^m\in U## or is it just shorthand for "this can be proved by induction"? In other words, for...

Now, for ##v\in V##, ##(T+I)v=0\implies Tv=-v##. That is, the null space of ##T+I## is formed by eigenvectors of ##T## of eigenvalue ##-1##. By assumption, there are no such eigenvectors (since ##-1## is not an eigenvalue of ##T##). Hence, if ##(T-I)v \neq 0## then ##(T+I)(T-I)v\neq 0##...

Question: Let ##\sigma\in S_n## be a permutation and ##T_{\sigma}## be the matrix we obtain from ##I## by appling ##\sigma## on the raws of ##I## (I.e ##\sigma## acts on the rows of ##I##) . Then: 1. ##\det(T_{\sigma}) = sgn(\sigma) ## and 2. ##T_{\sigma} T_{\tau} =T_{\sigma\circ \tau}##, for...

I've solved this problem using a fairly involved technique, where I compute the matrix ##e^{tA}## (the fundamental matrix of the system) with a method derived from the Cayley-Hamilton's theorem. It is a cool method that I believe always works, but it can be a lot of work sometimes. It involves...

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...

...this element ##r## can only be ##0## correct? The zero ring has only one element which is ##0##.

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!