Linked Questions

12
votes
4answers
3k views

History of Dual Spaces and Linear Functionals

Does anyone know or can anyone give a reference explaining how the concepts of a linear functional and particularly that of a dual space developed? I know Riesz published his famous representation ...
11
votes
3answers
1k views

Why are the four fundamental subspaces fundamental?

The four fundamental subspaces in linear algebra, as discussed by Gilbert Strang [1], are the kernel, image, dual space kernel, and dual space image (nullspace, column space, left nullspace, row space)...
7
votes
2answers
931 views

Dual space and covectors: force, work and energy

I am an engineer trying to understand the mathematical definitions and physical significance of vectors, dual vectors, and dual spaces. I understand how we take dot products and the covector "eats" ...
4
votes
2answers
993 views

Understanding purpose of dual vector space

I am a beginner in differential geometry and have questions/confusion about dual vector space. I took a look at this and this questions. But both did not resolve my question. We have standard ...
2
votes
1answer
1k views

understanding dual vectors and dual spaces

please help me with number example to understand better what is a co-vector and what is a dual vector space,as i know it is related to fact,that it eats vector,which means that it takes vector and ...
3
votes
1answer
581 views

Duality, Symmetry, Dual Spaces

The following is from the book "Tensor Methods in Statistics", which could be downloaded there http://www.stat.uchicago.edu/~pmcc/tensorbook/ I have a question regarding section 0.3.1, titled "...
3
votes
2answers
263 views

Why is it okay to think of naturally isomorphic sets as being equal to each-other?

Okay first of all here's what I understand of the two notions — A natural Isomorphism between two sets is an isomorphism that can be constructed without any choices involved, like choosing a basis or ...
1
vote
1answer
617 views

Understanding the dual space

I'm currently taking an honors course in linear algebra and am trying to get a feel for what the dual space actually is and what the motivation behind creating such a space is. From what I understand, ...
1
vote
2answers
234 views

dual formula to Bernoulli polynomials

$$ \tilde{B}_n(x) = \frac{(-1)^{n + 1}}{n!} \left( \delta^{(n - 1)}(x - 1) - \delta^{(n - 1)}(x) \right) $$ Wikipedia says this formulae is DUAL to the Bernoulli POlynomials but dual in what sense ?? ...
2
votes
1answer
250 views

Why do we care about transposes/duals of linear maps?

I'm just starting to go deeper in my linear algebra education, and this question: Why do we care about dual spaces? helped my intuition and motivation immensely, especially Matt E's answer. So, I'm ...
1
vote
1answer
524 views

Trying to understand dual space / dual vectors / 1-forms

I am new to this differential geometry business. I am trying to understand the concept of the dual space and dual vectors. I found this great answer on another stack exchange post here, but I have a ...
2
votes
1answer
92 views

Question about coordinates in linear functionals

In "Finite dimensional vector spaces" by Halmos (page 23) I read at some point: The concept of dual space was defined without any reference to coordinate systems. Also in an older thread here at ...
-1
votes
1answer
136 views

How to define the dual polytope for a given set of points in $\mathbf{R}^n$?

Let $S$ be a set of points in $\mathbf{R}^n$: $$S=\{ (x_1,\dots,x_n) \ | \ x_i \in \mathbf{R} \ ; i=1,\dots, n \}$$ How do I define and find the dual space or dual polytope for the set $S$.
2
votes
1answer
51 views

Why do we care about the general outer product $\mathbf u\otimes\mathbf v=\mathbf u\mathbf v^{\sf T}$?

Why do we care about dot and cross products? Why do we care about covectors? [essentially is the meaning of the question] Why do we care about dual spaces? So, why do we care about the general outer ...
1
vote
1answer
67 views

Example of the determination of a dual space

I would like a concrete example of the determination of a dual space. How to frame the example is up to you, but if you wish me to frame it, then consider the space spanned by the columns of \begin{...

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