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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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 ...
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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'= ...
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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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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 $AB-BA$. ...
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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
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 ...
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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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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
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\{ ...
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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
109 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
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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 ...
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2answers
50 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
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 $$ Tf(k)=[f(k)+f(n+1-k)].$$ ...
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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 ...
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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 ...
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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 ...
2
votes
1answer
44 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
83 views

Lights out variation proofs?

I would like some help solving these questions regarding a specific variation of the lights out game where all lights are initially off. The game can be played here (by double clicking edit then ...
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. ...
2
votes
3answers
976 views

Prove for minimum scalar product

The minimum scalar product of two set of data is when they are ordered in an inverse way. $$A=\langle 200, 8, 110, 300\rangle$$ $$B=\langle 22, 34, 88, 1 \rangle$$ Ordering both in an inverse way ...
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\\ ...
2
votes
1answer
874 views

Cosine Similarity between two sets of vectors?

I have words represented as vectors, and so I can compare two words using the cosine similarity of each word vector. But, now I'd like to extrapolate that and compare two sentences, each being a set ...
1
vote
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)= ...
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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.
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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
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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& ...
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 ...
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1answer
41 views
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?
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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 ...
0
votes
3answers
80 views

Dimension of $K[x]/x^{2}$ as a vector space

Let $K$ be a field, and $R=K[x]$ the polynomial ring over K. Let $J$ be the ideal generated by $X^{2}$ Show that $R/J$ is a K-space. What is its dimension? I know that the torsion submodule of $R/J$ ...
1
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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
2
votes
1answer
50 views

Find a basis for a space of functions

Let $X = \lbrace x_1, x_2, . . .,x_n \rbrace$. Find a basis for the space $\mbox{Map}(X,\mathbb R)$. Note: $Map(X,R)$ is the space of all functions that goes from $X$ to $\mathbb R$. Note 2: I was ...
0
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0answers
16 views

Prove the polynomials (1-t)… generate the polynomials space 3 degree

Prove the polynomials $\left(1-t\right)^{3},\,\left(1-t\right)^{2},\,1-t,\,1$ generate the $\mathsf{P\mathrm{_{3}\left(t\right)}}$ space of polynomials of degree $\leq3$ . I work this problem as ...
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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
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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 ...
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 ...
0
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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?
4
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1answer
83 views

Extreme points of intersection of the orthant (quadrant) with an Hyperplane in finite dimension vectorial space

Note : Having spent some time over the original problem below, I saw that it can be boiled down to a simpler problem. Here is that simpler problem : In a vectorial space (over $\mathbb{R}$) 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 ...
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0answers
10 views

$ \mathcal L_1 $-property inherited through normalization?

simple question regarding functional spaces. Assume $ u(t) $ is a signal defined for $ t \geq 0 $, with $ |u| \notin \mathcal L_1 $, and construct the signal $$ g(t) = \dfrac{|u(t)|}{\sqrt{1 + ...
2
votes
0answers
27 views

Perpendicular Vectors in 3D space

I was wondering whether given two Vector's v0 and v1 whether I could find the two perpendicular vectors at a given distance, d, from v1, perpendicular to the v0/v1 line. I know that v0 and v1 will ...
0
votes
1answer
25 views

Show that the linear operator is zero.

let $V$ be a finite dimensional vector space over $\mathbb C$ .Let $T:V \to V$ be a diagonalizable operator on $V$ such that $T$ acts nilpotently on some $y$ i.e. $T^m(y)=0$ for some $m \in \mathbb ...
0
votes
1answer
27 views

Find a basis and the dimension of the solution space W of the following homogeneous system [closed]

Good morning, I need help with this problem. Find a basis and the dimension of the solution space $W $of the following homogeneous system $\begin{cases} x+2y-2z+2s-t=0\\ x+2y-z+3s-2t=0\\ ...