0
votes
1answer
22 views

Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos
0
votes
3answers
73 views

High dimensional vector space references

Is there any good text book or review papers that introduce high dimensional vector spaces and its peculiarities as compared to generic/low-dimensional vector spaces? For example, high dimensional ...
3
votes
2answers
61 views

Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however: A standard definition of linear function (or linear map) is: $$f(x+y)=f(x)+f(y),$$ ...
0
votes
2answers
29 views

Splitting a matrix $A \in \mathbb{M}^{n \times n}(\mathbb{C})$by solving $Av = \lambda C v$ for some chosen $C$

If we know a matrix $A \in \mathbb{M}^{n \times n}(\mathbb{C})$ and solve $Av = \lambda v$ where we try to find $\lambda,v$, we can rewrite $A$ in a nice way. What if we choose a matrix $C$ and we ...
3
votes
2answers
53 views

Any suggestion on how to justify true/false question in linear algebra exams?

I have hard time bringing words on paper when it comes to true false justification of linear algebra problems. My technique is to use counter example for false and use book theorems for true ones. ...
2
votes
2answers
115 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
1
vote
1answer
88 views

Where to post discovered formulae? [closed]

I have discovered an alternate formula for the Fibonacci sequence and I would like to find a way in which I can present this. Please could you give me suggestions on how I can go about posting this ...
5
votes
0answers
84 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
1
vote
2answers
121 views

Linear Algebra without Matrices

How far could one get in linear algebra without matrices? It seems like the more I learn, the less I actually use them, but most of the basic theorems and invariants that learned first -- and still ...
0
votes
1answer
22 views

Definition of linear subspace

Let $k<d\in\mathbb{N}$. Given the following definition: $G=\{ S: S\text{ is }k\text{-dimensinal subspace of }\mathbb{R}^d\}$ Would you understand that $G$ contains only "homogeneous linear ...
5
votes
1answer
93 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
3
votes
0answers
73 views

Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
1
vote
3answers
62 views

Inner Product Spaces, suggestion for book.

Can you suggest me name of some books which would help me visualize IPS better? Like, books having diagrams and stuff?
13
votes
8answers
514 views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
0
votes
5answers
123 views

Ok, I know what does linear independence mean but why should I care?

I understand that for a set of vectors to be linearly independent, none of the vectors in the set should be a linear combination of some other vectors in that set. But why on earth should I care about ...
2
votes
3answers
97 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
1
vote
1answer
38 views

Do zeros present along the diagonal yield complex eigenvalues?

I was told today by a friend that having a zero along there main diagonal of a matrix will promote complex eigenvalues. I do not believe this is true because the below matrix Z has a zero present ...
8
votes
1answer
284 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
37 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
53 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
88 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 ...
12
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
14 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
44 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
307 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
20 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
35 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
458 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
23 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
87 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
269 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
218 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
141 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 ...
7
votes
3answers
460 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
40 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
349 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
98 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
191 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 ...
13
votes
4answers
346 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
498 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
442 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 ...
13
votes
10answers
429 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
3k 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?
9
votes
3answers
449 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
190 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
59 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
57 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 ...
5
votes
2answers
132 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
148 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: ...
27
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 ...