The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.
The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.
Questions tagged [dual-spaces]
1318 questions
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How can a vector be represented by a dual basis if a dual basis has a different form?
I am currently going through a book, "Tensor Algebra and Tensor Analysis for Engineers" by Itskov, and I am trying to understand what is meant by dual basis for a vector space. The author states that an arbitrary vector in a Euclidean space can be…
user3113647
- 302
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Rank of a matrix constructed by a dual space.
Let $V$ be a vector space of a finite dimension.
Let $ B = \left \{v_1,....v_n \right \} \subseteq V$, such that $\text{dim}(\text{Span}(B)) = k$
Let $C = (\lambda_1,....\lambda_n ) \subseteq V^{*}$ be a sequence of arbitrary linear…
GoodWilly
- 905
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why is the dual space defined the way it is?
The dual space $V^*$ of a vector space $V$ over $F$ is defined as $V\to F$.
This seems like a weird definition. In my limited experience with linear algebra, I've thought of the dual vector of a vector as the row vector version of that (column)…
user56834
- 12,925
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Covectors, Metric Tensors and Unit Vectors
In orthogonal cartesian coordinates $c\cdot c=|c|^2$ is the square of length of the vector $c$.
In oblique coordinates the square of the length of $c = c * g * c$ where g is the metric tensor and $g*c$ is a covector.
If $e_x$ and $e_y$ are the basis…
R. Emery
- 599
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Dual space kernel equality
I don't know how to show this one:
given elements $f,g ∈ V^∗$ of the dual space. I should now show that if $g = λf$ ,with $λ ∈ K\setminus\{0\}$ and $f$ is not $0$ ,
it follows that $\ker(f) = \ker(g)$.
Can Ener
- 15
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Two different kinds of dual of a vector
So there is a dual that converts vectors to bivectors and scalars to pseudoscalars (by multiplying by the pseudoscalar).
There is also a dual that converts vectors to covectors (one-forms).
Am I correct in assuming that these are two completely…
R. Emery
- 599
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A counter example of a proof for the independence of dual basis
Below is a proof for the independence of dual basis I found in Linear Algebra Done Righ (3rd edition, page 102):
Suppose $v_1,...,v_n$ is a basis of $V$. Let $f_1,...f_n \in V'$, which is the dual space of $V$, such that $f_i(v_j)= 1$ if $i = j$ and…
Thuc Hoang
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find the dual basis of $\mathbb R_{\leq2}[x]$ given a basis $B=(x^2-1,x^2-x,x^2+x)$
find the dual basis of $\mathbb R_{\leq2}[x]$ given a basis $B=(x^2-1,x^2-x,x^2+x)$
I know what a dual basis means, however I seem to miss something. here's my attempt:
I need to find a basis $B^\ast=(\phi_1,\phi_2,\phi_3)\, $ such that…
giorgioh
- 322
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How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$
How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$
I know the "regular" basis $B=\{ (3,1,0), (2,0,-1)\}$. But what is the dual basis?
user1068636
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Dual of the space of bounded functions on the unit hypercube
Which is the dual of the space $\ell^\infty([0,1]^d)$? Does the dual of such space consists of a set of functionals $\phi$ admitting the representation
$$
\phi(f)=\int_0^1 f\circ g(u) \psi_\phi(u) du, \quad f \in \ell^\infty([0,1]^d),
$$
with…
Jack London
- 1,756
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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{equation}V = \left[\begin{array}{rr}1 & 0 \\ 0 & 1…
thb
- 449
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2 answers
Finding the dual of linear map
Given $F: \mathbb{R}^2\to \mathbb{R}^3$ by $F(x,y)=(y,x,x+y)$, I have calculated its dual map $F':\mathbb{R}^3\to \mathbb{R}^2$ as $F'(x,y,z)=(x+y+z,x+z)$ by using its matrix representation w.r.t the standard dual basis. Is there any way to deduce…
kyborg
- 49
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Evaluation map infinite dimensional vector space
In a finite-dimensional vector space, we have an isomorphism ev: $ V \to V^{**}$, where ev(v)$: V^* \to F$ and $f \mapsto f(v)$. But what exactly happens in the infinite-dimensional case? Why is ev$(v)$ not surjective?
David Kwak
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Is what's a vector and covector subjective?
From what I understand, if you have $v \cdot w$ (or in general $v\otimes w=k$ where k is an element of a field), then either v dot or dot w can be considered as a linear functional and therefore a covector, and the other a vector. I've heard of a…
Pineapple Fish
- 1,512
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Show that $V/U\cong (U^0)'$
Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $F$. Show that $V/U\cong (U^0)'$.
We have that $U^0=\{f\in V' \ | \ \forall u\in U: f(u)=0\}$ is the annihilator and $V/U=\{v+U \ | \ v \in V\}$ is the quotient space.…
A.M.
- 3,944