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
30 views

How to prove if there exist unique $u$ and $w$ such that for any $v$, $v=u+w$, then $V$ is the direct sum of $U$ and $W$

How do I prove the statement: if there exist unique $u$ and $w$ such that for any $v$, $v=u+w$, then $V$ is the direct sum of $U$ and $W$? ($U,W,V$ are vector spaces, $u \in U, w \in W, v \in V$) I ...
0
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1answer
45 views

Basis and dimensions example

Every basis of $\mathbb R^6$ can not be reduced to a basis of $5$-dimensional subspace of $\mathbb R^6$ by removing one vector . Can anyone give an example for that?
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1answer
10 views

Show that $\int r\cdot n ds$ equals three time the volume of $\omega$.

Let $\Omega$ be an open region in $\mathbb{R}^3$ with surface $∂\Omega$ on every point $P$ of which the unit outward pointing normal $n = n(P)$ is well defined and smoothly varying. Let $r = (x, y, ...
2
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1answer
41 views

What do you call this equivalence relation? $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$

If $A, B$ are square matrices with coefficients in some ring, we say that $A$ is similar to $B$ if $A = PBP^{-1}$ for some invertible matrix $P$. Similar matrices represent the same linear operator ...
0
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1answer
37 views

Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines.

Given two lines in Cartesian form, find the vector equation of a line which passes through the intersection of two lines (and is perpendicular to both). No points given just the two equations. What ...
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1answer
83 views

Null space of matrix A and column space of transpose matrix A

Let A be an m×n matrix. Show that every vector v $\in R\ {^n} $can be written uniquely as w + u, where w is in the null space of A and u is in the column space of transpose A
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0answers
27 views

Symplectic form on $\mathbb R ^{2n}$

What are all symplectic form $\omega$ on $\mathbb R^{2n}$. Where, a ''symplectic bilinear form'' on $\mathbb R^{2n}$ is . a bilinear form: a map $\omega: \mathbb R^{2n}\times \mathbb R^{2n}\to ...
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1answer
22 views

Difference between $\mathbb{R}^4$ and $\mathbb{C}^4$ in subspace spanned by some vectors.

This is a problem in Hoffman / Kunze, Linear Algebra: Let $$\alpha_1=(1,1,-2,1), \quad \alpha_2=(3,0,4,-1), \quad \alpha_3=(-1,2,5,2).$$ Let $$\alpha=(4,-5,9,-7), \quad \beta=(3,1,-4,4), \quad ...
0
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1answer
32 views

Dimension and sum of linear subspaces

How to find the dimension of the intersection and the sum of linear subspace defined as linear span of the vectors systems. Vectors: $$ \begin{align} V_1&=\langle (1,1,0,1,1), ...
3
votes
1answer
83 views

Isomorphism between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
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1answer
32 views

Prove the image of basis elements is linearly independent

I was wondering if someone could give me a quick proof or counterexample to the following statement. Let $f:V \rightarrow W$ be a linear map between finite dimensional vector spaces $V$ and $W$, both ...
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1answer
33 views

Matrices for transformations

How can I find the matrices of part b and c? the answers are:
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1answer
21 views

vector spaces and euclidean n space

What's the difference between a real vector space and a euclidean $n$-space ?Are there any ? Both are denoted by $\mathbb{R}^n$ but we need $10$ axioms to define a vector space, but not the $n$-space. ...
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0answers
16 views

nondegenerate bilinear form $\mathrm{dim}{S}+\mathrm{dim}{S^{\perp}}=n$ [duplicate]

I was told that in a linear space $V$ with nondegenerate bilinear form$\langle\cdot,\cdot\rangle$ , and $S$ is a subspace of $V$. we have $$ \mathrm{dim}{(S)}+\mathrm{dim}{(S^{\perp})}=n $$ where ...
0
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1answer
23 views

Finding basis for column space of matrix

To find a basis for the column space of a matrix one finds the RREF of the matrix. The columns in the RREF are not a basis for the column space, but the same columns in the original matrix are a ...
1
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0answers
23 views

Vector on a sphere

I have for some time tried to understand the math behind explained in this post, but seem to not grasp. I think the way i visualize it might be incorrect, which make harder for me to grasp what is ...
1
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0answers
26 views

Do monomials form a basis for the vector space of real analytic functions?

Does the set ${1, x, x^2...}$ form a basis for the vector space of real analytic functions over the real numbers? It seems obvious that they span, but not obvious that they are independent.
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0answers
24 views

Exist this vectorial equality, and this is correct?

doing a problem about distance with vectors appears this identity: ...
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0answers
11 views

Is this curl operator with surface normal and tangential components valid?

Is this curl operator valid? $\nabla \times \mathbf{A} = (\partial_{\tau_1} A_{\tau_2} - \partial_{\tau_2} A_{\tau_1}) \hat{\mathbf{n}} - (\partial_n A_{\tau_2} - \partial_{\tau_2} A_n) ...
0
votes
1answer
14 views

Weird transposing after dot product and transformation

I'm reading a paragraph in a book where a plane equation ($N\cdot Q + D = 0$, N being the normal and D the distance from the origin, Q any point which belongs to the plane) is transformed by a matrix ...
0
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1answer
34 views

isomorphism from one vector space to another one

This is from my textbook I don't quite understand what isomorphism means. Greek word "isomorphism" means same structure, but how can we say $P_3$ has the same structure as $R^4$?
2
votes
1answer
35 views

Linear algebra over $\mathbb{Z}$

Suppose I have $v_1,\ldots,v_n$ vectors in $\mathbb{Z}^n$. Let $M$ be the matrix whose columns are $v_1,\ldots,v_n$. I would like to know if, as it happens with a vector space over a field, $M$ is ...
0
votes
2answers
28 views

What does the product <basis vector times the underlying field> represent?

I am confronted with the following definition: Let $K$ be a field and $e_1,e_2,\ldots,e_n$ the standard basis of the $K$ vector space $K^n$. For $1\leq i\leq n$ let ...
2
votes
1answer
51 views

Weird characteristic polynomial question

Let $F_A:\,\mathrm{M}_2(\mathbb{C})\to\mathrm{M}_2(\mathbb{C})$ be defined by $\mathrm{M}\mapsto \mathrm{MA}+\mathrm{AM}$. I am doing a question which asks me to write the characteristic polynomial of ...
1
vote
1answer
25 views

Unit base vectors in a new coordinate system

Let's assume we have a function $f:\Omega =R^2 \rightarrow R $ $f(x,y)=x+2xy+x^2y$. Obviously our unit base vectors on $\Omega$ are $e_x=\hat{i}$ and $e_y=\hat{j}$. Now we want to change the ...
3
votes
1answer
29 views

Geometric intuition of the equation of a plane

Let $\pi$ be a plane in an $d$-dimensional space with normal vector $\underline{w} = [w_1, \dots,w_d]^T$. If point $\underline{p} = [p_1, \dots,p_d]^T$ is in the plane and $\underline{x}= = [x_1, ...
1
vote
2answers
18 views

Inverse result for Direct sum of vector space theorem

From direct sum of vector space, we know that given a vector space $V$ and subspaces $U$ and W, if $V= U+W$, and $U\cap W = \{0\}$, then $V= U \oplus W$. My question is given a vector space $V$ which ...
0
votes
2answers
66 views

Basis for intersecting subspaces - is there a trick here?

I'm doing this problem, which gives me these subspaces of $\mathbb{R}^4$ $$U=\text{span}\left\{\;\begin{pmatrix} 3\\ 2\\4 \\ -1\end{pmatrix},\;\begin{pmatrix} 1\\ 2\\1 \\ ...
-1
votes
2answers
23 views

Cartesian Equations Intersecting

One line $L_1$ has a cartesian equation $x+1=\frac{y}{3}=-z.$ Another line $L_2$ has a cartesian equation $2x+1=2y+1=z+a$, where $a$ is not known. $L_1$ and $L_2$ intersect in a point, so find the ...
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2answers
28 views

Is polynomials of pair degree a vector space?

How can we prove the above statement? additionnaly , if we take only polynomials with monomials of pair degree can we conclude the same, is it a subspace of $K_n[X]$?
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1answer
14 views

How can i prove a closed ball is the closure of a open ball?

do I stick to definitions or theres a helpful theorem arround? Let $W:=\{y\in X : ||y-x||\leq r\}$ and $S:=\{y\in X : ||y-x||<r\}$ for any $r>0$. If $z\in W$ and $z_n:=(r-1/n)z$ with $n\in ...
0
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0answers
43 views

A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
0
votes
1answer
19 views

Sequence Space Basis

Let $V$ be the sequence space of all sequences $a=(a_0,a_1,\ldots)$ that are eventually zero, that is for every $a$ there is a $N$ such that $a_n=0$ for every $n>N$. My question is: how can I ...
4
votes
2answers
77 views

How do I link dimension of a normed vector space with closedness?

Let $X$ be a Normed Vector Space, for any $x\in X$ and $r>0$. Let $W:=\{y\in X : \|y-x\|\leq r\}$ and $S:=\{y\in X : \|y-x\|<r\}$ Prove: $W$ is closed if $\dim(X)<\infty$ I can't think of a ...
1
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2answers
34 views

Problem in solving a question of vector space.

The question is : Let, $V$ be the subspace of all real $n \times n$ matrices such that the entries in every row add up to zero and the entries in every column also add up to zero. What is the ...
0
votes
1answer
27 views

Constructing representation of $G$

Say we are given an arbitrary group $G$ and an arbitrary vector space $V$ over some field. How can we construct a representation of $G$ on some vector space from this data? Initially I wanted to ...
1
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2answers
24 views

why any vector can be wriiten as the sum of two components in the row space and nullspace?

My textbook says that: there is a $m\times n$ matrix A, any vector x in $R^n$ can be written as the sum of a component $x_r$, in the row space, and a component $x_n$ in the nullspace: $$x=x_r+x_n$$ ...
1
vote
2answers
32 views

why nullspace is the largest subspace perpendicular to the row space?

The proof from my textbook is "If x were a vector orthogonal to the row space, but not in the nullspace, then the dimension of $C(A^T)^\perp$ would be at least n — r + 1. But this would be too large ...
5
votes
1answer
62 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
6
votes
2answers
81 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
1
vote
1answer
23 views

Which of the following is a vector subspace of $R^3$

To prove that $F$ is a K-vector subspace of $ E$ it suffices to prove $\alpha f_1+\beta f_2 \in F$ with $(\alpha, \beta)$ $\in K²$ and $f_1 , f_2 \in F$ . For trivial cases and easy ones it seems ...
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0answers
25 views

what is the field over “ $K^0$ ”?

What is " $K^0$ " ? And does it have a Basis ? I considered the field over K with the power of zero to be a field with zero dimension so that it cannot have a basis.. is it right or am I wrong?
0
votes
1answer
14 views

Expression related to dual norm on bounded linear functionals

Given a vector space $V$ and a norm $\|\cdot\|$ on $V$, the dual norm $\|\cdot\|^*$ on $V^*$ is given by $\|f\|^* = \sup \left\{\frac{f(v)}{\|v\|}\right\}$ over all nonzero vectors $v$. I've found ...
-2
votes
0answers
26 views

Check if a subset is a subspace of some Vector space

For $V= \{F: \Bbb R \to \Bbb R\}$ and $S=\{f \in V \mid f(2)=f(3)\}$, how do I go about proving $S$ is a subspace of $V$? I think it is but cannot figure out a way to prove it as usual... $f(2) = ...
0
votes
1answer
34 views

Cut Space of Vertices without Orthogonal Complement of Cycle Space?

I am studying sparse graphs where their complements tend to be dense (not sparse). I understand this so that the sparse graph has a sparse adjacency matrix while its graph complement is not most ...
0
votes
0answers
13 views

What is cut space of directed graph (digraph)?

A cut is partition of vertices into two disjoint subsets. Digraph is a directed graph. Cut space is defined for an undirected graph as by Wikipedia where the definition for an undirected graph, ...
4
votes
3answers
221 views

Proof: Sum of dimension of orthogonal complement and vector subspace

Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is defined as $$ W^\perp= \left\{ v ...
0
votes
2answers
42 views

Dimension of a vector space of polynomials in 3 variables of degree $d$ over $\mathbb{R}$ [duplicate]

Let $V$ be a vector space of homogeneous polynomials in 3 variables $x_1, x_2$ and $x_3$ over $\mathbb{R}$. What is $\dim V$? I think it will be some expression in terms of $d$ but I am not ...
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0answers
29 views

What is the dimension of $\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}$?

I have $$\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}.$$ Here $f$ is the following endomorphism $$f(P) = (x^2-x+1)P(-1)+(x^3-x)P(0)+(x^3+x^2+1)P(1),$$ where $P\in\mathbb{R}_{n}[x]$. My ...
0
votes
2answers
24 views

Basis in the space of polynomials

Do the vectors $\mathbf{p}_1(x)= 2+x+4x^2$, $\mathbf{p}_2(x)= 1-x+3x^2$ and $\mathbf{p}_3(x)= 3+2x+5x^2$ make a basis in the space of polynomials of degree at most $2$? If "yes", expand the ...