For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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2
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3answers
52 views

Is there a linear operation such that $F(1,1,1) = (1,2,3),F(1,2,3) = (1,4,9),F(2,3,4) = (1,8,27)$?

The exercise asks me verify if there exists a linear operator $F$ such that: $$F(1,1,1) = (1,2,3)\\F(1,2,3) = (1,4,9)\\F(2,3,4) = (1,8,27)$$ First I tried to write a vector $(x,y,z)$ as a linear ...
1
vote
2answers
38 views

Finding basis and dimension based on definition of space

I've got two vector spaces $U$ and $V$ over division ring $\mathbb{T}$ . Space $W$ over division ring $\mathbb{T}$ is defined as $W =\{( u, v ); u \in U, v \in V \}$ with operations $(u_1, v_1) + ...
3
votes
2answers
61 views

Is $U \subseteq V$ for given $U$ and $V$?

How can I decide if it is true, that $U \subseteq V$ for given $$ U = \operatorname{span}\{(−1,−1,−2,2,1),(3,2,3,1,−3),(1,0,−1,5,−1),(1,−2,−3,3,1)\}\\ V=\{(x_1, x_2, x_3, x_4, x_5) \in ...
3
votes
2answers
163 views

How to visualize the gradient as a one-form?

I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the ...
4
votes
0answers
200 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
1
vote
2answers
60 views

Find a basis for the subspace

How can I find a basis for the subspace $V:=\{v = (v_1, v_2, \cdots , v_n) \in R^n: v_1+v_2+ \cdots +v_n=0\}$ of $R^n$ for any $n$? I know that I must show that the basis is linearly independent and ...
2
votes
1answer
131 views

Inverse gradient as line integral in Mathematica

I found a nice paper about inverse vector operators here. I have successfully defined a Mathematica function for inverse curl and inverse divergence, however I can't figure out how to do inverse ...
2
votes
2answers
100 views

The determinant of a complex linear operator regarded as a real linear operator?

I was trying to solve the following question Let $T: \mathbb{C}^{2} \rightarrow \mathbb{C}^{2}$ a linear operator with determinant a+bi. If we regard $\mathbb{C}^{2}$ as a vector space over ...
0
votes
2answers
35 views

Linear algebra, Linear Transformations

Let $V$ be a finite-dimensional vector space over the field $F$ and let $W$ be a subspace of $V$. If $f$ is a linear functional on $W$, prove that there is a linear functional $g$ on V such that ...
0
votes
1answer
30 views

linear independance of a span

I might be going in circles, so would appreciate some clear input....As i understand, a span of a set of vectors can include dependent vectors. A span 'creates' a subspace, therefore a subspace can ...
2
votes
3answers
90 views

Find a basis for the subspace $\left\{\begin{bmatrix}x & y \\ z & t\end{bmatrix}, x-y-z = 0\right\}$

The exercise gives me the subspace $$\left\{\begin{bmatrix}x & y \\ z & t\end{bmatrix}, x-y-z = 0\right\}$$ and ask me to show that these two sets are basis for this subspace: $$B = ...
2
votes
1answer
75 views

Classification of finite dimensional inner product spaces.

Given a complex inner product on a finite-dimensional vector space, is there always a matrix $M$ such that $\langle x,y \rangle=y^*Mx$. What are the properties of such a matrix? I saw on the wiki ...
-1
votes
2answers
166 views

Differences of $V_1 \cup V_2$ and $V_1 +V_2$? [duplicate]

Let $V_1,V_2$ are subspaces of vector space $V$ . Differences of $V_1 \cup V_2$ and $V_1 +V_2$ ?
0
votes
5answers
199 views

What is the formal mathematical representation of a “force”?

In mechanics, it is usual to represent a force by a 3-vector. When it is necessary to consider the turning effect of a force, the 3-vector is commonly "attached" to a point on its line of action. In ...
0
votes
2answers
36 views

The number of real values for which set is Not a basis of $R^{2} $

Given set is $(a, a^{2}) , (a^{2} , a ) $ . As i see that for a =1 ,0 and -1 set is not a basis .But how do i check it mathematically , if there are any other values ? Thanks
0
votes
2answers
991 views

Change of basis matrix for polynomials?

I've understood what a change of basis matrix is, and how it's structured. So a change of basis matrix from $B$ to $C$ is the matrix $M$ such that: $${\begin{bmatrix} &\\ \\ \\\end{bmatrix}}_B ...
1
vote
1answer
37 views

Prove that $\det(\text{Id}+T)\ge 1+\det(T)$

Let a self-adjoint operator $T:V\to V$ above $\mathbb{C}$, such that $\langle Tv,v \rangle \ge 0$ (so it's essentially a real number). We have learned before that for this kind of $T$, all it's ...
0
votes
1answer
112 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
1
vote
2answers
1k views

How to prove a type of functions is a subspace of the vector space of all functions.

I've been working on algebra and want to know how to determine if a certain type of function is a subspace of the vector space $\mathbb{R} \to \mathbb{R}$. So far I've been using the two properties ...
2
votes
2answers
295 views

hypothetical 4 dimensional vector space

Before I start asking the question, I want to apologize for my illiteracy in latex maths and the abstraction of my question. My question is, is it possible for me to define a hypothetical 4 ...
0
votes
2answers
150 views

Automorphism group of vector space

I was trying to understand definition of representation and trivial representation thus came across the case where $ V= K $ here $V$ is a vector space over a field $K$ and thus $Aut_K (V) \cong ...
1
vote
0answers
30 views

To find dimension of subspace

Let W be subspace spanned by$ (2i,0,1,2i) , (0,2i-2,i-3,0), (-i,1,0,i), (1,1,1,1) $ I have tried to reduce it to RREF , but its such a pain . Is thereany shorter way i could do this ...
1
vote
1answer
36 views

Find equation for an isomorphism such that

In $\mathbb{R}^4$ plane V is given, $V=span(\alpha_1,\alpha_2)$ where $\alpha_1=[1,3,4,1]$, $\alpha_2=[1,2,2,3] $ a) Find the formula for isomorphism $\varphi:\mathbb{R}^4\rightarrow\mathbb{R}^4$, ...
2
votes
1answer
60 views

Change of basis with a nonlinear operator

Given a vector space $V$ and its two basis: $\mathcal{B}$ given by vectors $\{e_i\}$ and $\mathcal{B}'$ given by vectors $\{e'_i\}$, why are the two basis necessarily connected by a linear ...
0
votes
0answers
25 views

About hermiticity implying non-nilpotency for infinite dimensional vector space(Corrected)

Suppose $A$ is a non-trivial linear operator acting upon infinity dimensional vector space. Say given $A^2$=0 and provide that $A$ is hermitian. Is this sufficient to conclude $A$ is non nilpotent? I ...
0
votes
1answer
52 views

To determine basis for $V \cap W$

Let $V$ be vector subspace of$ R^{4} $ spanned by vectors $( 1,1,1,-1)$ and $(1,-1,0,1)$. Let $W$ be another vector subspace of $R^{4}$ spanned by $(1,1,-1,1)$ and $(1,3,4,-5)$. Determine basis of ...
1
vote
1answer
48 views

Prove that a set of matrices is a linear space

Prove that the set of matrices $$v:=\left\{ \begin{pmatrix} 2x-y+z & x-2y-2z \\ x+y-z & 3x+y+2z \end{pmatrix} \middle|\, x,y,z \in R\right\}$$ Is a linear space above $R$ and find it's base. ...
0
votes
2answers
39 views

Prove two simple projection statements

Let $U$ be a subspace of $\mathbb{C}^n$ and suppose $v \in \mathbb C^n$. Let p be the projection of the vector $v$ onto the subspace $U$. That is, p is defined as $P_U(v)$. Prove : (1) $\langle ...
0
votes
2answers
96 views

Why study real vector spaces of dimension $n$ other than $\Bbb{R}^{n}$?

Every vector space of dimension $n$ is isomorphic to $\mathbb{R}^{n}$. Why do we study other (finite-dimensional, real) vector spaces?
0
votes
0answers
68 views

Minkowski sum and difference do not cancel

I need to prove that $ ( A\oplus B )\ominus$ $B$ and $( A\ominus B)\oplus$ $B$ need not equal $A$ for all sets $A$, $B$,where $\oplus$ and $\ominus$ denote the Minkowski sum and difference. As ...
1
vote
1answer
58 views

Schur's Triangularization Lemma in Hefferon's Linear Algebra textbook

I'm reviewing some material and came to this: Fix a basis $B = \{\vec{\beta}_1, \ldots, \vec{\beta}_n\}$ for $V$ ($V$ is a vector space) and observe that the spans $$ [\emptyset] = \{\vec{0}\} ...
0
votes
1answer
482 views

How to express the Pythons' NumPy linspace or arange arrays mathematically?

How one can express digital one dimensional array, such as x = np.linspace(0, 10, 1000) or x = np.arange(-1, 1, 0.01) (examples ...
2
votes
1answer
71 views

Does there exist an infinite dimensional vector space over an infinite ordered field which cannot have any inner-product imposed on it?

The title says it all. I'm wondering if there is any infinite dimensional vector space over some infinite ordered field such that we cannot impose any inner product on it at all. I understand that ...
0
votes
1answer
514 views

Can i Find the Matrix from Eigenvalues and Eigenvectors?

If i given eigenvector: $$V_1=\begin{pmatrix} {1\over \sqrt{3}}\\{1\over \sqrt{3}}\\{1\over \sqrt{3}}\end{pmatrix} , V_2=\begin{pmatrix} {1\over \sqrt{6}}\\{-2\over \sqrt{6}}\\{1\over ...
0
votes
1answer
47 views

what is the geometical interpretation of $\vec a.\vec b$? [duplicate]

what is the geometical interpretation of $\vec a.\vec b$?(dot product) I know the projection of $\vec a $ on $\vec b$ is $\vec a.\hat b$. But what is a projection here?
1
vote
1answer
25 views

Qestion about Eigenvector, basis for the solution

I'm confused with some question currently I'm trying to solve. If you help that will be grateful. Given the matrix find eigenvalues and eigenvectors $$ A = \begin{bmatrix} 4 & -2 ...
1
vote
1answer
41 views

Solution space of a Differential Equation

Generally, initial conditions to an $n^{th}$ order ODE involve initial conditions only involving derivatives up to the degree $ n-1 \ (like \ y^{(n-1)}(0) \ = \ A).$ Even a basis of the space of ...
0
votes
0answers
62 views

What makes a norm-Gaussian inner product space “infinite-dimensional”?

Suppose we define an $\mathbb{R}^m$ inner product space in which the inner product of $\mathbf{x}$ and $\mathbf{y}$ is $\exp\left(-\|\mathbf{x} - \mathbf{y}\|\right)$. In PCA and machine learning, we ...
2
votes
1answer
163 views

Show that $V^*$, set of all Linear Transformations from $V$ to $R$, is a vector space

$V$ is a vector space, and $V^*$ is the set of all LT's from $V$ to $\mathbb{R}$. a) Show that $V^*$ is a vector space. b) Suppose $\{v_1,\dots,v_n\}$ is a basis for $V$. For $i = 1,\dots ,n$ define ...
0
votes
0answers
52 views

Prove that there exist $W$ such that $V=V_1\oplus W=V_2\oplus W$

Let $V$ be a finite-dimensional vector space. If $V_1$ and $V_2$ are distinct linear subspaces of $V$ such that $\dim V_1=\dim V_2$, show that there exists a linear subspace $W$ of $V$ such that ...
0
votes
1answer
67 views

Extreme points of complex sphere of dimension n in 1-norm.

I came up with the following question while learning about different norms in $\mathbb{C}^n$. For $z=(z_1, \ldots, z_n)^T \in \mathbb{C}^n$ we consider the 1-norm: $\|z\|_1= \sum_{k=1}^n|z_k|$. Let ...
3
votes
1answer
330 views

Confusing notation $D(p)(x)$ in a vector space of polynomials

If we have a vector space that consists of all polynomials of degree less than or equal to 4, and we consider the following function: $$D(p)(x) = 2.5\cdot p(x-1)$$ where $p$ is a function from the ...
0
votes
3answers
60 views

Are there any simple/explicit examples of a finite vector space?

By finite vector space, I mean a non-trivial vector space with a finite number of elements, not just a finite field. I'm hoping for a really simple example, even better if that set is explicitly ...
2
votes
1answer
33 views

Base of vector space from a finite set [closed]

Let $V$ be a vector space of finite dimension. $S=\{v_1,...,v_r\} \subset V$ and $Span(S)=V$. For each $v_i\in S$ there is a linear combination from $S\setminus \{v_i\}$. How can I show that for each ...
1
vote
3answers
110 views

To determine Nullity of $T$

Let $V$ be vector space of polynomials of degree $\leq n$ . And $ T : V \rightarrow \mathbb R ^{m}$ be defined as $T (P (x)) = (P (1) , P (2) ,..., P (m) )$ I have to determine nullity of $T$ . ...
1
vote
1answer
51 views

Finding a basis and the dimension of $W_1\cap W_2$

Suppose $W_1,W_2$ are subspaces of $\mathbb{R}^4$. $W_1$ is spanned by $(1,2,3,4), (2,1,1,2)$ and $W_2$ is spanned by $(1,0,1,0),(3,0,1,0)$. I have to find a basis for $W_1\cap W_2$. I have ...
0
votes
1answer
77 views

Dual basis in a finite separable extension

I am reading the book Algebras, Rings and Modules, volume 1, by M. Hazewinkel and at the page 193 there is a proof about why the integral closure of a ring in a separable finite extension L over $k$ ...
1
vote
2answers
60 views

Modules isomorphism

Studying vector spaces, we can find the well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
3
votes
0answers
52 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
0
votes
1answer
44 views

A real vector space with a complex structure is naturally a complex vector space

I am struggling with this exercise from the book 'Tensors and Manifolds: With Application to Physics', by Robert H. Wasserman: Corresponding to each $a \in \mathbb{K}$ there is a linear operator ...