All Questions
Tagged with big-list linear-algebra
18 questions
9
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3
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Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
16
votes
2
answers
1k
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Definitions of determinant by unique features
A well-known definition of the determinant is:
The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized.
See e....
37
votes
17
answers
13k
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Listing applications of the SVD
The SVD (singular value decomposition) is taught in many linear algebra courses. It's taken for granted that it's important. I have helped teach a linear algebra course before, and I feel like I need ...
34
votes
8
answers
4k
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Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
5
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0
answers
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Where have you encountered "arrangement spaces"?
I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).
Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
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41
votes
4
answers
6k
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Linear algebra in terms of abstract nonsense?
The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.
I was wondering what portions of basic linear algebra (first couple of courses) fall ...
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19
votes
5
answers
1k
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List of counting proofs instead of linear algebra method in combinatorics
I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
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8
votes
4
answers
2k
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NP-hard problems in linear algebra and real analysis [closed]
I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...
0
votes
1
answer
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Applications of the natural bilinear forms on the direct sum between a vector space and its dual
As is known, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew-symmetric bilinear forms
$$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 (\xi(Y) \pm \eta(X)).$$
I am interested in ...
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19
votes
3
answers
2k
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Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
56
votes
21
answers
18k
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Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
109
votes
19
answers
38k
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Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
51
votes
22
answers
19k
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Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
19
votes
17
answers
7k
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Vector spaces without natural bases
Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural".
To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
10
votes
4
answers
8k
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Any reference on multilinear algebra [closed]
Do you know any good reference on multilinear algebra?
56
votes
21
answers
14k
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Linear algebra proofs in combinatorics?
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
62
votes
25
answers
70k
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Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
368
votes
31
answers
80k
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Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...