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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24 views

How to denote dimensions

I am struggling with nomenclature. If I have matrix $M \in \mathbb{R}^2 \times \mathbb{R}^4$ it would be considered an element of an 8-dimensonal vector space. If I index $M$ by two indices $i$ and ...
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0answers
25 views

Linear Algebra.

I have the exercise: Calculate the coordinates of the point $M = (m_1, m_2, m_3)$, such that $\frac{d(M,P)}{d(M,Q)}=\frac{1}{2}$ where $P = (5, 8, 1)$ and $Q = (4, 2, 2)$, here $d(A,B)$ denotes the ...
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28 views

Primordial elements of a vector space

We were given the following problem in our Algebra class. Let $V$ be a $K$-vector space (not necessarily finite dimensional), and fix a basis $(e_i)_{i \in I}$ of $V$. If $x = \sum \xi_ie_i \in V $, ...
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2answers
71 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
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2answers
23 views

Set of linear transformations being a vector space

Let $V$ and $W$ be vector spaces, $T$, $T_1$, and $T_2$ linear transformations from $V$ to $W$, and $k$ a scalar. We define new transformations $T_1 + T_2$ and $kT$ by the formulas: $$(T_1 + ...
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1answer
77 views
+50

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
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1answer
26 views

Determinant of the matrix representation of an isomorphic linear transformation

Are there any theorems or special properties about the determinant of a matrix representation of an isomorphic linear transformation?
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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
19 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 ...
5
votes
2answers
165 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. ...
1
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1answer
14 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
vote
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 ...
4
votes
2answers
71 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}, ...
1
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1answer
23 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
30 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 ...
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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
30 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 ...
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1answer
26 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
19 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
37 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?
0
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1answer
16 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 ...
0
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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
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1answer
84 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
48 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
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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
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2answers
46 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 } $ $$0\cdot\vec{v}=\vec{0} $$ ...
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0answers
28 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
27 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
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2answers
67 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
20 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
26 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
45 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
20 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
26 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
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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
27 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
19 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
39 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
votes
1answer
41 views

Subspace proof/dimension of..

How can I do the following question?
-1
votes
1answer
41 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
122 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
55 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?