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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1answer
41 views

Linear algebra proving question, matrix algebra [closed]

I am stuck on this question and I don't even know how to start.Attached below is the picture of the question
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4answers
46 views

Prove that $F(x,y) =(2x+y, x+4y)$ is bijective

I need to prove that $$F(x,y) =(2x+y, x+4y)$$ is injective and surjective. I started by assuming $$F(x_1,y_1) = F(x_2,y_2)$$ or $$(2x_1+y_1, x_1+4y_1) = (2x_2+y_2, x_2+4y_2)\implies\\2x_1+y_1 = ...
2
votes
3answers
49 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
20 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
52 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
69 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 ...
3
votes
0answers
36 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 ...
0
votes
0answers
16 views

Transformation on vec operator to reorder elements - a bit like sudoku.

I have the following expression $ \begin{pmatrix} vec(Y_1) \\ vec(Y_2) \\ \end{pmatrix}$, where $Y_i = [y_i,~x^1_i, x^2_i]$. Each element in $Y_i$ is a column vector. Then let $Y_i$ be a $T ...
1
vote
2answers
39 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 ...
1
vote
1answer
51 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
51 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
30 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
19 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
38 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 = ...
1
vote
1answer
27 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
82 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 $V_1 +V_2$ ?
1
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0answers
44 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
29 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
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2answers
54 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
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1answer
35 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
21 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
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2answers
88 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
159 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 ...
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0answers
20 views

what exactly is a center invariant subspace?

I'm studying robust control, and I have a matrix with two invariant subspaces. One is stable, which I assume is spanned from the eigenvectors with real part less than zero, and the other is a center ...
0
votes
2answers
31 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 ...
0
votes
1answer
32 views

Linear Algebra Subspaces questions [closed]

Let V be a vector space over a field F and M,N≤V. Let M be the plane x+y+z=0 and N be the line x=y=z in R^3 . Show that R^3=M+N . (M+N is direct sum)
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0answers
26 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
25 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
23 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
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0answers
22 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
37 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
40 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
35 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
68 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?
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0answers
34 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 ...
0
votes
0answers
14 views

How to determine a supplementary subspace?

Problem: Determine a supplementary subspace $U$ to the subspace $W = span\left\{(1,6,4)\right\}$ of $(\mathbb{R}, \mathbb{R}^3, +)$. Determine also for the vectors $v = (2,2,0)$ and $w = (0,4,4)$ ...
1
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1answer
31 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
28 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
34 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
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1answer
55 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
41 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
18 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
29 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
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0answers
40 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
71 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
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0answers
41 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
22 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
182 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
46 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
31 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 ...