2
votes
1answer
67 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
1
vote
0answers
31 views

Analogy between linear basis and prime factoring

I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and ...
0
votes
1answer
45 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
1
vote
2answers
84 views

Why is the $O$ (zero) matrix important?

In reading my linear algebra book I found it quite interesting that they made the following comment: One important property of addition of real numbers is that the number $0$ is the additive ...
11
votes
5answers
2k views

Why teach linear algebra before abstract algebra?

Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...
0
votes
0answers
11 views

Transformation Matrix of an Endomorphism

I am going through my script at the moment and there is an example given which I do not understand: Let $F \in \text{End}(\mathbb{R}^2)$ be represented by $ \begin{pmatrix} 0 & 1 \\ 1 & 0 ...
1
vote
0answers
39 views

Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
8
votes
4answers
285 views

Linear algebra questions that a high-schooler could explore

Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?
2
votes
0answers
17 views

Motivation behind Binet form and its generalization to arbitrary coefficients

Binet form (http://www.proofwiki.org/wiki/Binet_Form) gives a closed-form solution to $n^{th}$ term in a series with recurrence relation as below $U_{n}=m\cdot U_{m-1}+U_{m-2}$ I have two questions ...
2
votes
1answer
26 views

Soft Question about Mobius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Mobious transformation matrix as a change of basis for C?
7
votes
6answers
407 views

Thinking of mathematics in terms of analogs

I think the way that I've come to think about mathematics is becoming problematic and I'm wondering if I should abandon it. When I study mathematics, I find myself trying to compare the mathematical ...
1
vote
0answers
21 views

Mathematical standard term for a function of (different) operator arguments

In quantum mechanics, one often considers functions of linear operators, like $$f(A,B) = A\cdot B + e^A \cdot B^2$$ where $A,B$ are linear operators. In physics this often causes confusions, as some ...
2
votes
0answers
64 views

James Munkres Elementary Linear Algebra

Can anyone who has the book give a quick opinion? There are no reviews in any of the websites, and what's weird is the book is freaking expensive, but only 42 pages.Isn't this weird?I know rigorous ...
4
votes
1answer
260 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
8
votes
1answer
183 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
0
votes
1answer
107 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...
8
votes
3answers
257 views

What does the sign $\propto$ mean?

I am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name. So, I would like ...
1
vote
1answer
35 views

Complement Theorem and Steinitz Theorem, are the same thing?

There is a theorem about basis of a vector space that says: Let $u_{1}, u_{2},...,u_{p}$ be linear independent vectors of some vector space, $E$. Suppose that each $u_{i}$, with $i=\{1,2,3,...p\}$ ...
1
vote
0answers
201 views

Difficulty in learning calculus 3 years after one has learned calculus 2?

Academically speaking, I'm a late bloomer. Long story short, I was a bad kid, teachers told me I wasn't good at math and I believed them. Years later, as an almost 30 year old man, I started going ...
4
votes
0answers
86 views

definition of determinant in Artin

In Artin, the discussion on determinant starts from the standard recursive expansion by minors. Artin defines determinant as a function $\delta$ from a square matrix to a real number. Then Artin lists ...
2
votes
2answers
160 views

Fundamental problem of Linear Algebra

What is the fundamental problem of linear algebra? I understand it is a big question and not easy to explain completely, and seems no way to prove an answer is correct. I just wanna listen to you ...
12
votes
4answers
272 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
1
vote
4answers
275 views

Book Recommendations for Linear Algebra Proofs

I'm taking a graduate Linear Algebra course and have limited experience writing proofs (mostly from a discrete math class). Can anyone recommend good books to teach you how to write proofs for linear ...
1
vote
2answers
287 views

Most important Linear Algebra theorems?

I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem: A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix. This is ...
10
votes
8answers
298 views

$2\times2$ matrices are not big enough

Olga Tausky-Todd had once said that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." There are, however, assertions about matrices that are true for ...
1
vote
2answers
2k views

What are the applications of matrices in real world?

Matrices are considered very important in mathematics. What are some examples of applications of matrices to real world problems that would be understandable by a layman?
4
votes
0answers
109 views

Proof for '$AB = I$ then $BA = I$' without Motivation?

I have read this question page (If $AB = I$ then $BA = I$) by Dilawar and saw that most of proofs are using the fact that the algebra of matrices and linear operators are isomorphic. But from a ...
1
vote
2answers
151 views

Jordan Normal Form and eigenvalue 0

I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question: Why is it that when you ...
5
votes
1answer
54 views

What is the underlying structure that makes this analogy so good?

In "Linear Algebra Done Right", the author draws (in my opinion) a fantastic parallel between $\mathbb{C}$ and $\mathcal{L}(V)$ (where $V$ is an $\mathbb{F}$-inner product space). In this analogy, he ...
0
votes
1answer
52 views

Norm, Euclidean Space and Distance

To a complete layman, how would you define the following terms intuitively? $norm$ , $euclidean$ $space$ , and $euclidean$ $distance$ ? Note: I have tagged Linear Algebra and Probability Theory ...
4
votes
2answers
115 views

Why are quadratic forms so special and why not investigate in higher forms?

Ok, this is a soft question. If $K$ is a field of characteristic different from $2$, one can use the polarization identity to get a one-to-one correspondence between homogeneous polynomials of ...
2
votes
1answer
129 views

Book Recommendation for Second Course in Linear Algebra

I only took a non-rigorous linear algebra course (It was designed for non-math students). I finished most of Hungerford's algebra. Now I have two choices to study more advanced linear algebra: ...
26
votes
6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
1
vote
3answers
106 views

Rating changes and probability calculations for chess world championship

I have some interesting questions that have to do with the rating changes and calculations for the Anand-Carlsen world championship. Most of this has to do with solving a system of linear equations. ...
25
votes
2answers
935 views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
2
votes
1answer
188 views

4 big ideas in algebra that have rich connections to other fields? [closed]

In his controversial post criticizing high school algebra, Grant Wiggins issued a challenge to his readers: Can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of ...
3
votes
1answer
167 views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
7
votes
1answer
146 views

Motivation for Jordan Canonical Form

I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I ...
3
votes
3answers
109 views

Basis of a basis

I'm having troubles to understand the concept of coordinates in Linear Algebra. Let me give an example: Consider the following basis of $\mathbb R^2$: $S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and ...
5
votes
2answers
294 views

Advice: Modern vs. Classics

First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
4
votes
2answers
175 views

Why is Matrix Multiplication Not Defined Like This? [duplicate]

I'm sure everyone already thought about this at least one time. Why matrix multiplication is not defined the way showed below? $$\left( \begin{array}{ccc} a_{11} & a_{12} & \ldots \\ a_{21} ...
7
votes
4answers
1k views

Importance of eigenvalues

I know how to find eigenvalues and eigenvector .But I dont know what to do with that. What is there use? Can anyone explain me that?
4
votes
4answers
4k views

How does linear algebra help with computer science

I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult ...
2
votes
0answers
131 views

Properties of Non-Diagonally Dominant Matrix

I have a question about properties of matrices which are or are not diagonally dominant. So I understand that a diagonally dominant Hermitian matrix with non negative diagonal entries is positive ...
11
votes
3answers
212 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
4
votes
2answers
105 views

Books for linear algebra over commutative rings

I was thinking about reviewing linear algebra to recover many theorems that I can use over commutative rings with unity. But it seems very tedious and I did not want to make any mistakes on these ...
3
votes
1answer
97 views

Motivation for the double dual

I was reading on the double dual of a vector space $V$ recently. I was wondering what applications (within mathematics) there are for this concept and/or what was the motivation for the development of ...
3
votes
1answer
78 views

Presentation, Reduction and Generalization in Mathematics: The Case of Linear Algebra

Apologies for the grandiose title, but it is motivated by a serious consideration. Linear algebra, LA hereafter, is an enormously interesting area of mathematics. What's more, it is fairly ...
2
votes
0answers
106 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
6
votes
3answers
1k views

Advice on Understanding Vector Spaces and Subspaces

currently I am studying Vector spaces and sub spaces. I enjoyed working with matrices and using the Gaussian-Jordon elimination and I also had no problems with cofactor expansion and determinants in ...