# What is Linear functionals: Definition and 22 Discussions

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted Hom(V, k), or, when the field k is understood,

V

{\displaystyle V^{*}}
; other notations are also used, such as

V

{\displaystyle V'}
,

V

#

{\displaystyle V^{\#}}
or

V

.

{\displaystyle V^{\vee }.}
When vectors are represented by column vectors (as it is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).

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1. ### MHB Need some hints on my HW about Linear functionals

I not very good at using the LaTex editor, so I took a photo of my HW questions. For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would \varphi (P) be when m=1? For the second question, I know that a linear functional have two...
2. ### Proving a mapping from Hom(V,V) to Hom(V*,V*) is isomorphic

Homework Statement Let V be of finite dimension. Show that the mapping T→Tt is an isomorphism from Hom(V,V) onto Hom(V*,V*). (Here T is any linear operator on V). Homework Equations N/A The Attempt at a Solution Let us denote the mapping T→Tt with F(T). V if of finite dimension, say dim...
3. ### Proof regarding transpose mapping

Homework Statement Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1. Homework Equations N/A The Attempt at a Solution Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...
4. ### Annihilator of a Direct Sum: Proving V0=U0⊕W0 for V=U⊕W

Homework Statement Suppose V=U⊕W. Prove that V0=U0⊕W0. (V0= annihilator of V). Homework Equations (U+W)0=U0∩W0 The Attempt at a Solution Well, I don't see how this is possible. If V0=U0⊕W0, then U0∩W0={0}, and since (U+W)0=U0∩W0, it means (U+W)0={0}, but V=U⊕W, so V0={0}. I don't think this...
5. ### Proof regarding linear functionals

Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...
6. ### Linear functionals: Φ(u)=0 implies Φ(v)=0, then u=kv.

Homework Statement Suppose u,v ∈ V and that Φ(u)=0 implies Φ(v)=0 for all Φ ∈ V* (the duel space). Show that v=ku for some scalar k. Homework Equations N/A The Attempt at a Solution I've managed to solve the problem when V is of finite dimension by assuming u,v are linearly independent...
7. ### I A Question about Notation and Continuous Linear Functionals

I have reading through various sources on linear functionals, but all seem somewhat inconsistent with regard to denoting the set of all linear functionals and the set Also, what is the standard definition of a continuous linear functional? I really couldn't find much besides this Let ##f : V...
8. ### Linear Functionals & Inner Products: Is This Theorem True?

Is this "theorem" true? Relationship between linear functionals and inner products Suppose we have a finite dimensional inner product space V over the field F. We can define a map from V to F associated with every vector v as follows: \underline{v}:V\rightarrow \mathbb{F}, \ w \mapsto \langle...
9. ### Finding Dual Basis of Linear Functionals for a Given Basis in C^3

Hello, Problem, let B={a_1,a_2,a_3} be a basis for C^3 defined by a_1=(1,0,-1) a_2=(1,1,1) a_3=(2,2,0) Find the dual basis of B. My Solution. Let W_1 be the subspace generated by a_2=(1,1,1) a_3=(2,2,0), let's find W*, where W* is the set of linear anihilator of W_1. Consider the system...
10. ### Bi Linear Functionals and Symmetry

Homework Statement Show that ## \displaystyle B_1(u,v)=\int_a^b (p(x) u \cdot v + q(x) \frac{du}{dx} \cdot v)dx## is a bilinear functional and is NOT symmetric Homework Statement Bilinear relation ##B(\alpha u_1+\beta u_2,v)=\alpha B(u_1,v) +\beta B(u_2,v)## (1) ##B(u, \alpha v_1+...
11. ### Linear Functionals: Why Not ##I(u) = \int_a^b u\frac{du}{dx}dx##?

Homework Statement Why does this not qualify as a linear functional based on the relation ##l(\alpha u+\beta v)=\alpha l(u)+\beta l(v)##? ##\displaystyle I(u)=\int_a^b u \frac{du}{dx} dx## Homework Equations where ##\alpha## and ##\beta## are real numbers and ##u## , ##v## are...
12. ### Linear Functionals - Continuity and Boundedness

Homework Statement Prove that a continuous linear functional, f is bounded and vice versa. Homework Equations I know that the definition of a linear functional is: f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) and that a bounded linear functional satisfies: ||f(|x>)) ||...
13. ### Linear functionals on a normed vector space

I have a question: If x\in X is a normed vector space, X^* is the space of bounded linear functionals on X, and f(x) = 0 for every f\in X^*, is it true that x = 0? I'm reasonably confident this has to be the case, but why? The converse is obviously true, but I don't see why there couldn't be an...
14. ### Linear Functionals, Dual Spaces & Linear Transformations Between Them

I have a question about mappings that go from a vector space to the dual space, the notation is quite strange. A linear functional is just a linear map f : V → F. The dual space of V is the vector space L(V,F) = (V)*, i.e. the space of linear functionals, i.e. maps from V to F. L(V,F)=...
15. ### Tangent vectors as linear functionals on F(M)

Let M be an n-dimensional manifold, with tangent spaces TpM for each point p in M. Let F(M) be the vector space of smooth functions M --> R, over R, with the usual definitions of addition and scaling. Tangent vectors in TM can be defined as linear functionals on F(M) (Fecko: Differential...
16. ### If m<n prove that y_1, ,y_m are linear functionals

Homework Statement Prove that if m<n, and if y_1,\cdots,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j]=0 for j=1,\cdots,m. What does this result say about the solutions of linear equations? Homework Equations...
17. ### Math: Solving Linear Functionals w/ Riesz Representation

How do I solve this problem- I know it has something to do Riesz represenation but am having difficulty connecting dots Conside R4 with usual inner product. Find the linear funcitonal associated to the vector (1,1,2,2). What am I missing- is this problem complete or is there something...
18. ### Linear Functionals Inner Product

Assume that m<n and l_1,l_2,...,l_m are linear functionals on an n-dimensional vector space X . Prove there exists a nonzero vector x \epsilon X such that < x,l_j >=0 for 1 \leq j \leq m. What does this say about the solution of systems of linear equations?This implies l_j(x)...
19. ### Understanding Linear Functionals: Help Me w/ Example Problem!

I am studying for a final I have tomorrow in linear algebra, and I am still having trouble understanding linear functionals. Can someone help me out with this example problem, walk me through it so I can understand exactly what a linear functional is? Is the following a linear functional? \ y...
20. ### Understand Linear Functionals & Vector Space X

Here is the problem I have been asked to solve: Assume that m < n and l1, l2, . . . , lm are linear functionals on an n-dimensional vector space X. (a) Prove there exists a non-zero vector x in X such that the scalar product < x, lj >= 0 for 1 <= j <= m. What does this say about the solution of...
21. ### Linear Functionals and Operators: Exploring Properties and Relationships

1) let S:U->V T:V->W be linear operators, show that: (ToS)^t=S^toT^t. 2) let T:V->U be linear and u belongs to U, show that u belongs to Im(T) or that there exist \phi\inV* such that T^{t}(\phi)=0 and \phi(u)=1 about the first question here what i tried to do: (ToS)^{t}(\phi(v))=\phi...
22. ### Solve Ly=y''(x)+4xy'(x)-2x for Linear Functionals

I'm not quite sure if this is a linear functional but the question asks: if L=D^2+4xD-2x and y(x)=2x-4e^{5x} I am to find Ly=? My first impressions to solve this is the take Ly=y''(x)+4xy'(x)-2x i'm not quite sure how to solve this but I got: y''(x)=-100e^{5x} y'(x)=-20e^{5x}+2...