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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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
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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
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
45 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
43 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
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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
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
54 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 ...
1
vote
1answer
85 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 ...
0
votes
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 ...
2
votes
1answer
28 views

Confusion on Differential Operators and Notation

My teacher wrote the following on the board: $\frac{d}{dx}: C^1(a,b)\rightarrow C^0(a,b)$ I thought that a 1st order differential operator takes a function which is continuous and maps it to some ...
0
votes
0answers
14 views

Union of the unit circle $S^1$ and the curve is connected but not path-connected

Prove that the union of the unit circle $S^1$ and the curve $W=\{(x, y) \in \mathbb{R^2} | x=(1-e^{-t})cost, y=(1-e^{-t})sint, t \geq 0\}$ is connected but not path-connected A connected space is ...
0
votes
1answer
28 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\\ ...
-1
votes
1answer
27 views

Find coordinate vector in matrix vector space

How do I do this question? I don't understand the notation that describes B what is the superscript ij? what is E?
0
votes
1answer
32 views

Show that $V(R)$ is an $\mathbb{R}$-vector space

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Show that $$ V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
1
vote
0answers
20 views

Is a finite dimensional vector also a covector?

I am reading something where it seems they have defined a covector/linear functional as a map $F: \mathbb{R}^n \to \mathbb{R}$ where $F$ is a vector For example, $F(x) = \begin{bmatrix} 1 & 2 ...
0
votes
2answers
58 views

Show that $u$ is a multiple of $v$ or vice versa in the following

Show that if $\left | \left \langle u,v \right \rangle \right | = \left\|u \right \| \left\|v \right \|$ then either $u$ is a multiple of $v$ or $v$ is a multiple of $u$. Here is what I did: Assume ...
1
vote
0answers
29 views

Showing Linear dependency

Show that if each of the vectors $\left\{v_1, v_2, . . . , v_n\right\}$ is a linear combination of the vectors $\left\{w_1, w_2, . . . , w_n\right\}$, then $\left\{v_1, v_2, . . . , v_n\right\}$ is ...
3
votes
1answer
13 views

Vector Space external direct sum

Question: Give an example to show that it is possible for $A \oplus B = A\oplus B'$ without having $B=B'$, where $A,B,B'$ are subspaces of $_FV$ I really can't imagine this, say let $A \oplus B = ...
0
votes
0answers
11 views

Spanning Spaces by Different Basis

I have a query related to spanning space by two bases $S_1=\{V_1+V_2, V_3, V_1-V_4,V_3-V_2\}$ $S_2=\{V_1, V_2, V_3, V_4\}$ Can we consider spaces generated by $S_1$ and $S_2$ to be equivalent?? Or ...
1
vote
0answers
23 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
0
votes
1answer
29 views

Orthogonal complement of orthogonal complement of U equals U

Let $V$ be of finite dimension over some field $F$. Let $ξ\in T_2(v)$ symetric bilinear form. Let $U\subset V$. suppose $ξ|_U$ is not degenerate, is it necessarily $(U^⊥)^⊥=U$ ? my approach: I ...
0
votes
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 ...
0
votes
0answers
23 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes ...
0
votes
1answer
25 views

Forming a basis of P3(R) from a set S.

I seem to have a good understanding of spanning sets and linear independence which then becomes essential for understanding basis, but I am unsure how all this works for the field of polynomials. I ...
2
votes
2answers
22 views

Vector representation

I saw many people representing vectors like this: -----> in a 2D plane. Why do you need the little arrow head over there in the end? Doesn't it make that a ray ...
0
votes
2answers
28 views

proving n linearly independent vectors are generating

Given n linearly independent vectors $v_1, v_2, ... , v_n$ in a real vector space $V$ where $\mbox{dim}(V)=n$, how do I prove the vectors generate $V$? I understand why this is true, I can visualize ...
0
votes
0answers
22 views

In a vector space every subspace of codimension $d$ is the zero set of $d$ linear functionals

Let $V$ be a vector space over a field $k$ (not necessarily finite dimensional), and let $W$ be a subspace of codimension $d$, i.e. $\dim(V/W) = d$. Is it true that there exists $d$ linearly ...
0
votes
1answer
58 views

Determine a subspace in R^5

I'm having trouble with this question: For (c) obviously I can prove that by the "Closed under scalar multiplication" But the other two I am a bit confused. Hope someone might shed some light on ...
0
votes
0answers
25 views

Showing that common p-Norm isn't a norm anymore for $0\lt p\lt 1$

I have the following problem: I need to show that the common p-Norm defined as: $$||.||_p: (v_1, \dots ,v_n) \to (\sum^n_{i=i} |v_i|^p)^{1/p}$$ doesn't constitute a norm on $\Bbb R^n$ for $n \ge 2$ ...
1
vote
2answers
38 views

Expanding a Proof of Induction on $\Bbb N $ to $\Bbb Q $ (Linear Algebra)

My problem is the following: I have an $\Bbb R$ Vectorspace called $V$ and had to show via induction that $\langle nv, w \rangle=n \langle v, w \rangle$ for $v,w \in V$ and $ n\in \Bbb N$. (it's not ...
0
votes
1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
1
vote
2answers
23 views

Find a basis of the subspace of polynomial subspace

Find a basis of the subspace of $${\mathbb R}^3$$ defined by the equation $$-9 x_1 + 3 x_2 + 2 x_3 = 0$$ I'm looking on how to approach this problem since my instructor only showed us how to prove if ...
2
votes
1answer
21 views

How to prove that $T(x)=[x]_a$ is surjective?

Let $V$ be a vector space and let $\alpha=\{v_1,\ldots,v_n\}$ be a basis for $V$. Let $T:\ V\to \Bbb{R}^n$ defined by $T(x)=[x]_a$ for every $x\in V$. How to show that $T$ is surjective (onto). Here ...
1
vote
1answer
38 views

Geometric intuition for finite vector spaces?

There is a powerful geometric intuition for real vector spaces. Is there any good way of visualizing vector spaces over finite fields?
0
votes
0answers
9 views

An open ball of a normed vector space is path-connected?

I tried to proove that an open ball of a normed vector space is path-connected. Consider $B(x,\epsilon)$ an open ball of a normed vector space $E$. Take a point $b\in B$. The path connects $x$ and $b$ ...
0
votes
1answer
22 views

The Existence of $n-1$ Dimensional Linear Subspace

Let $K$ be an infinite field, $V$ is an $n$ dimensional($n>1$) vector space over $K$. $\alpha_1,\alpha_2,\cdots,\alpha_s \in V$ are non-zero vectors. Proof there exists an $n-1$ dimensional ...
0
votes
2answers
24 views

find a scalar $\alpha$ that satisfy the following

when $v\neq 0$, find a scalar $\alpha$ such that $z:=u-\alpha v$ satisfies $\left \langle z,v \right \rangle = 0$ Is there some trick to this? I tried solving this explicitly and I just ended up ...
1
vote
1answer
31 views

Solve the following system of differential equation

Let $A = (a_{ij}\in M_3(R))$ be a real matrix and let $P:= \begin{pmatrix} 0 &1 &0 \\ 0 &1 &1 \\ 1 &1 &0 \end{pmatrix}$ such that $P^{-1}AP = \begin{pmatrix} 1 &0 ...