# 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.

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1. ### A gardener collected 17 apples...

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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
3. ### Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1

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...
4. ### Prove that range ##T'## = ##(\text{null}\ T)^0##

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...
5. ### Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.

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...
6. ### Given specific v, dimension of subspace of L(V, W) where Tv=0?

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...
7. ### Challenge Math Challenge Thread (October 2023)

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8. ### I Transfer rank2 tensor to a new basis

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9. ### I Dimension and solution to matrix-vector product

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...
10. ### Decompose 4x4 determinant into 24 determinants -- How many are zero?

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11. ### Identifying matrices as REF, RREF, or neither

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...
12. ### I Row space, Column space, Null space & Left null space

\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...
13. ### B Question on basic linear algebra (new to the subject)

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...
14. ### I Using Linear Algebra to discover unknown Forces

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...
15. ### I Understanding the operation in ##(\mathbb{z_6})^{*}##

...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...
16. ### I Zero Element in a Ring: The 0 Ring Has Only One Element

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17. ### B Commutative diagram conventions

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...
18. ### I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'

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...
19. ### Classical Source recommendation on Differential Geometry

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...
20. ### Challenge Math Challenge - June 2023

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...
21. ### Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

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!
22. ### Find the solutions of the system, for all λ

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23. ### I Find the Eigenvalues and eigenvectors of 3x3 matrix

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...
24. ### Proving eigenvalues of a 2 x 2 square matrix

For this, Does someone please know why the equation highlighted not be true if ##(A - 2I_2)## dose not have an inverse? Many thanks!
25. ### Using inverse to find eigenvalues

For this, I don't understand how if ##(A - 2I_2)^{-1}## has an inverse then the next line is true. Many thanks!
26. ### Eigenvalue of a matrix

For this, I am confused by the second line. Does someone please know how it can it be true since the matrix dose not have an inverse. Many thanks!
27. ### I Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

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...
28. ### Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

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...
29. ### Intro to Linear Algebra - Nullspace of Rank 1 Matrix

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...
30. ### I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

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31. ### I If T is diagonalizable then is restriction operator diagonalizable?

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...
32. ### A The eigenvalue power method for quantum problems

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...
33. ### If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

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...
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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...
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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...
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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...
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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...
38. ### I Intuition for why linear algebra is needed in quantum physics

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...
39. ### Spectral decomposition of 4x4 matrix

## A = \pmatrix{ -4 & -3 & 3 & 3 \\ -3 & -4 & 3 & 3 \\ -6 & -3 & 5 & 3 \\ -3 & -6 & 3 & 5 } ## over ## \mathbb{R}##. Let ## T_A: \mathbb{R}^4 \to \mathbb{R}^4 ## be defined as ## T_A v = Av ##. Thus, ## T_A ## represents ## A ## in the standard basis, meaning ## [ T_A]_{e} = A ##. I've...
40. ### Expectation of Product of three RVs

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}?
41. ### I The Price of Beer - Linear Algebra Problem

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...
42. ### I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

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...
43. ### Show a function is linear

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...
44. ### Intro Math Are the (translated) High School Japanese maths textbooks by Kodaira Good?

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/...
45. ### Linear algebra problem with a probable typo

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...
46. ### Prove relation between the group of integers and a subgroup

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47. ### I Proof about pre-images of functions

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...
48. ### Prove the identity matrix is unique

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...
49. ### Linear operator in 2x2 complex vector space

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...
50. ### Linear Transformation from R3 to R3

"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.