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
23 views

If a vector subspace is open, then it's the whole space

If $V'$ is a subspace of $V$ and it is open then $V=V'$. I've seen a similar question about this, but they talk about "non-empty interiors", is non-empty intetior similar to talkin about open sets? ...
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1answer
26 views

I think im missing linear property in this normed vector space how should i approach?

Let $V$ be a normed vector space and $V'$ a subspace, $x\in V-\{0\}$show that a)If $\exists \eta > 0$, such that $\{y\in V :\space\space ||y||<\eta\}\subset V'$ then $\frac{\eta x}{||x||}\in ...
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0answers
11 views

Affine transformations and its decomposition

Let A and B be affine spaces with respective vector spaces V and W. A map $f$ from A to B is called an affine map if there exists a linear transformation $f'$ from V to W such that ...
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1answer
22 views

affine spaces, affine hyperplanes [closed]

I am totally confused with the definition of affine spaces and affine hyperplanes. Informally an affine subspace is a space obtained from a vector space by forgetting about the origin. Mathematically ...
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3answers
44 views

Shortest Distance between planes

This is a question which puzzled our entire math class including our teacher, I'm referring to part (b), we're fine with part (a). We don't understand the reason for taking the dot product and the ...
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2answers
174 views

Is there a way to cut a an ellipsoid with a plane such that it gives an circle?

I'm trying to answer this In $\Bbb {R^3} $ consider the ellipsoid: $2x^2+3y^2+4z^2=1$ It exists a subspace of dimension 2 which intersection with the ellipsoid is a circle. Justify any answer. ...
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1answer
17 views

Do automorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an automorphism, i.e. a bijective endomorphism. If $V$ is finite-dimensional, we know that the characteristic ...
1
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1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
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2answers
72 views

Dimension of the subspace of the polynomial ring over $\mathbb R$

Suppose $P_n =\{ f(x) \in \mathbb R[x] : \deg(f(x)) \leq n\}$ and $W = \{ p(x) \in P_n : p(x) = p(1-x) \}$. Find the dimension of subspace $W$. Firstly I am showing that $W$ is a ...
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2answers
33 views

Prove this is a metric, what else should I consider?

Let $C_b(\mathbb{R})$ be the space of the bounded continuous functions with values in $\mathbb{C}$ defined in $\mathbb{R}$ ($f:\mathbb{R}\rightarrow\mathbb{C}$) prove that: with $x\in \mathbb{R}, ...
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1answer
24 views

Finding out vectors that screw up linearly independence when given a set

I want to Find the vector space spanned by $A =$ {$(1,1,0,1),(1,2,-1,1),(3,4,-1,3),(-1,-3,-2,-1)$} By definition it's all the linear combinations I can make with those 4 vectors, However I ...
1
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2answers
32 views

Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero ...
0
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2answers
28 views

Can you give me a hint on this proof of a subspace of vectors?

$V=\{(x_n)\in l^2 | $It has only a finite number of vectors$ \neq 0\}$ prove V is a subspace of $l^2$ but it isn't closed. I have problems understanding what does $l ^2$ means, and what the sentence ...
2
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0answers
34 views

Proof $S$ is the intersection of $m$ subspace of $V$ of dimension $n-r$

Let $V$ be a $k$-space with $dim(V)=n$. Let $S\subset V$ be a subspace, $\dim\left(S\right)=k<n $. For each $r\in\mathbb{N}$ with $1\leq r\leq n-k$, prove that $S$ is the intersection of $m$ ...
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0answers
9 views

What is diemension of R(L) and N(L) and this L is one-0ne , onto?

Let $P_3$ be polynomial of degree at most 3 and $\\L:P_3\rightarrow P_3$ be a linear transformation by $L(p)=xp''(x)-4x^2p'(x)$ Then this linear transformation is one-one , onto ? and what is the ...
1
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1answer
27 views

$V$ be a vector space ; $f,g \in \mathcal L (V)$ ; $f\circ g-g \circ f=I$ ; then is the set $\{g^n: n\ge 0\}$ linearly independent?

Let $f,g$ be linear operators on a vector space $V$ such that $f\circ g-g \circ f=I$ , where $I$ is the identity operator on $V$ ; then is it true that the set $\{g^n: n\ge 0\}$ is linearly ...
1
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1answer
21 views

Transition matrix of polynomial.

Good night, i need help with this. Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\} $ I found a basis for W, $\left\{ ...
2
votes
1answer
40 views

Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
0
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1answer
10 views

how to prove non-pivot cloumns is the sum of preceding preceding pivot columns in RREF

For example, we can see that $V_{3}=2V_{1}-3V_{2}, V_{5}=2V_{1}-2V_{2}-V_{4}$, but how can we mathematically prove the theory behind it?
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1answer
19 views

row space of A is equal to the row space of rref?

This is a proof from a textbook What I don't undersdand is, clearly the cofficients for r_i is not equal, unless a_j is 0 (k has to be non-zero), but we want a_j to be any number, don't we? so a_i ...
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0answers
7 views

Exterior algebra subspace of all grade-n wedge products of a vector

Let $V$ be a finite-dimensional vector space, and let $\Lambda(V)$ be its exterior algebra. Then if $S_k = \text{span}(k_1,k_2,...,k_n)$ and $\hat k = k_1 \wedge k_2 \wedge ... \wedge k_n$, there is ...
3
votes
1answer
108 views

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an ...
4
votes
1answer
49 views

Proving that a set of functions is a linear subspace of a vector space

I am attempting to solve the following problem: Let $V$ be the vector space of all continuous functions $f : R → R$ with point-wise addition and scalar multiplication defined. (a) Show that $M_1$ = ...
3
votes
2answers
18 views

Linearity of the right inverse of a surjective linear map

Suppose we have a surjective linear map $f:V\to V$ on an infinite-dimensional vector space $V$. We know that every surjective map has at least one right inverse. So I was wondering... I know not all ...
2
votes
0answers
26 views

Is the union of dual cone and polar cone of a convex cone is a vector space?

There is an exercise in the book Matrix Algebra that ask to show if $C$ is a convex cone, then the union $C^* \cup C^0$ is a vector space. Where $C^*$ is dual cone and $C^0$ is polar cone of $C$. I ...
0
votes
2answers
47 views

Proof of $0 \cdot\vec{v}=\vec{0} $ for all $ \vec{v}$ in the vector space

According to this webpage from this Lemma 1.17, which states: In any vector space V , for any $\vec{v}\in V \text{ and } r \in\mathbb{R} ,\text{ we have } $ $$0\cdot\vec{v}=\vec{0} $$ ...
0
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0answers
29 views

Evaluation of complex integral?

I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following: group ...
2
votes
1answer
30 views

Find dimension of a Vector Space.

Let $E=\{1,2,\ldots,n\}$, where $n$ is odd. $V$ is the vector space of all functions mapping from $E$ to $\mathbb R^3$. Find $\dim(V)$. Consider $T:V\to V$ such that $$ Tf(k)=[f(k)+f(n+1-k)].$$ ...
0
votes
2answers
70 views

Calculate Rotation Matrix to align k n dimensional vectors

I have a $k$ number of $n$-dimensional vectors written with respect to two rotated frames: $X= \{\vec{x}_1,\vec{x}_2,...,\vec{x}_k\}$ and the same rotated vectors: $X'= ...
0
votes
1answer
21 views

Direct sum and $FG$ homomorphism

Let $V$ be an $FG$-module and suppose that $$V=U_{1} \oplus...\oplus U_{r}$$ Each $U_{i}$ is an $FG$-submodule of $V$. For $v=u_{1}+...+u_{r}\in V$ and $u_{i} \in U_{i}$ Define $\pi_{i}: V \to V ...
0
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0answers
27 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
0
votes
2answers
46 views

Solve a geometry problem by using vectors.

In triangle $ABC$, the bisector of angle $A$ meets side $BC$ in point $D $ and the bisector of angle $B$ meets side $AC$ in point $E$. Given that $DE$ is parallel to $AB$, show that $AE = BD$ and that ...
1
vote
1answer
23 views

Linearly independent sets of vectors

Find $3$ vectors $a$, $b$ and $c$ in $\mathbb{R}^3$ such that {$a$, $b$}, {$a$, $c$} and {$b$, $c$} are each linearly independent sets of vectors, but the set {$a$, $b$, $c$} is linearly dependent. ...
0
votes
2answers
29 views

Projection of vector onto span

Let $y = \begin{bmatrix}1\\2 \\3 \\4 \end{bmatrix}$ and $S=Span\left ( \begin{bmatrix}1\\ 1\\ 1\\ 1\end{bmatrix},\begin{bmatrix}0\\1\\ -1\\ 0\end{bmatrix},\begin{bmatrix}0\\ 1\\ 1\\ ...
0
votes
1answer
52 views

Dimension of differential equation - vector space

What is the dimension of this? and why? my guess is that it is 4 but I don't know how to show this.
1
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1answer
28 views

Why is the magnitude of the curl of a vectorfield twice the angular velocity?

(if V is a vectorfield describing the velocity of a fluid or body, and $x\in R^3$) I agree that it should be when you look at the calculation, but intuitively speeking... If $\nabla \times V(x)= ...
0
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0answers
31 views

For which $A$ is true: $tr(XAY)=tr(YAX)$

$n \in \mathbb N,\forall X,Y \in \mathbb K^{n \times n},A \in \mathbb K^{n \times n} $ For which A is true: $tr(XAY)=tr(YAX)$ My answer would be if A is the identity matrix, but is there something ...
0
votes
1answer
21 views

All kinds of Row Space of a matrix

For example, determine a basis for the row space of $$A=\begin{pmatrix} 1& -1& 1& 3& 2\\ 2& -1& 1& 5& 1\\ 3& -1& 1& 7& 0\\ 0& ...
2
votes
1answer
43 views

Finding linear transformation for vector space of matrices

Good evening everyone, I understand how to find a determinant. What does it mean to have a linear transformation from the space $V$ of $2\times 2$ upper triangular matrices to $V$. Also, how did ...
-1
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1answer
41 views

Subspace proof/dimension of..

How can I do the following question?
-1
votes
1answer
42 views

Prove that the infinite union of linearly independent sets is linearly independent

I'm trying to prove this: Let $X_1,X_2,...,X_n,...$ be linearly independent sets of a vector space a. If $X_1\subset X_2\subset X_3\subset ... \subset X_n\subset X_{n+1}\subset ...$, prove ...
1
vote
3answers
132 views

Let v = (1, 1, 1, 1). Find a basis for…

How can I do this? In particular, I do not understand u.v=0
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2answers
57 views

Vector Space span/basis {1,$sin^2(x),cos^2(x)$}

How can I determine whether or not it (a) spans the vector space provided (b) is a basis for this vector space?
1
vote
2answers
33 views

Prove or disprove the claim: $Ker(T^{n}) = Ker(T^{n+1})$ for infinite dimension

Let $T:V \rightarrow V$ , if $V$ is infinite dimensional, can one still claim that $Ker(T^{n}) = Ker(T^{n+1})$ for some $n \geq 1$ ? If yes prove it, if not provide a counter example. I feel that the ...
0
votes
1answer
29 views

Where do matrices of real numbers lie?

I have a question on the space where matrices of real numbers lie. Suppose I have a vector $x$ of real numbers with dimension $p\times 1$. I usually write $x\in \mathbb{R}^p$. Consider now a matrix ...
0
votes
1answer
29 views

Linearly Independent or Dependent

Prove or disprove the following: If a set $T=(x_{2}-x_{1}, x_{3}-x_{1}, ..., x_{k}-x_{1})$ is a set of linearly independent vectors, then $S=(x_{1},x_{2}, ..., x_{k} )$ is a set of linearly ...
0
votes
1answer
40 views

Are all vector spaces closed under addition and scalar multiplication? If so, why?

The definition of a vector space doesn't explicitly include closeness under addition and multiplication. Is there a proof that shows or disproves it?
1
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0answers
39 views

Basis of the space of homogeneous polynomials clarification

I want to prove the following proposition. Let $H(n,m)$ denote the vector space of homogeneous polynomials of degree $m$ in $n$ variables over $\mathbb{C}$. Then here exist a finite number of ...
2
votes
2answers
53 views

If $V$ is finite-dimensional with $J : V \to V$ such that $J^2 = -id$, then $V$ has even dimension

Let $V$ be a $\Bbb R$-vector space, with $J$ being an endomorphism $J: V \to V$ with $J^2=-id$ (identity). I already had to show that $V$ became a $\Bbb C$-vector space with the scalar ...
3
votes
1answer
79 views

What is the most general/abstract way to think about Tensors

In their most general and abstract definitions as Mathematical Objects : A Scalar is an element of a field used to define Vector Spaces A Vector is an element of a Vector Space. Since a Scalar ...