Tagged Questions

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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
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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 ...
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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: $$f(x+y)=f(x)+f(y),$$ ...
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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 ...
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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. ...
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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, ...), ...
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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 ...
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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 ...
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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 ...
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 ...
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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 ...
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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 ...
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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?
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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 ...
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“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 ...
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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 ...
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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 ...
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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 $i=\{1,2,3,...p\}$ ...
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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 ...
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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 ...
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 ...
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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 ...
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 ...
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 ...
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$2\times2$ 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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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 ...