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 ...
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)...
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" ...
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 ...
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 ...
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 "...
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 ...
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, ...
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 ?? ...
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 ...
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 ...
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 ...
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 $S$.