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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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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Understanding the dimension of a particular subspace in Linear Algebra in C³

I am having trouble to determine the dimension of the subspace of T. How is it done, when there is just one vector given?
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1answer
69 views

Getting an isomorphism from a short exact sequence of inner product spaces

Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
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Free graded k[x] modules have homogeneous bases

I was reading the article "Cary Webb. Decomposition of graded modules. Proceedings of the American Math- ematical Society, 94(4):565–571, 1985" where in the beginning "Free graded k[x] modules have ...
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3answers
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What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by describing ...
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1answer
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Can you construct a basis for an infinite dimensional vector space from a set of vectors that span that space?

Suppose I have an infinite dimension vector space V (not necessarily countably infinite). Suppose a have a set S that spans the space. If V is finite dimensional, it is trivial to construct a basis ...
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geometic description of vector span

i am having some poblem with this question Show that the set if vector is linearly dependant. If the vector span a line, provide the equation of the line. If the vector span a plane, provide the ...
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25 views

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$

Suppose we have three vectors $x,y,z\in \mathbb{R^3}$ Further suppose that x ⊥ y and y ⊥ z then is x ⊥ z?
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What does this symbol means in Banach's spaces?

Let $X,Y$ be non trivial vector spaces such that $L(X,Y)$ is Banach's $\Rightarrow Y$ is Banach's. I'm missing the definition of $L(X,Y)$ in my notes, also can you give me a hint? I'm new in Banach's ...
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Prove that this transformation inverse exists and it's bounded

If $X,Y$ are Normed Vectorial Spaces, $T$ is a bounded lineal transformation. Prove that if exists $b>0$ such that $\|Tx\|\geq b\|x\| \forall x\in X$. Then $T^{-1}:Y\rightarrow X$ exists and it's ...
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44 views

How to prove that 1, sin(x), xsin(x) and cos(x) are linearly independent?

My attempt: Suppose that $a\cdot 1 + b\cdot \sin(x) + c \cdot x\sin(x) + d\cdot \cos(x) = 0$. Let $r = \sqrt{b^2 + d^2}$ and $\tan \phi = \frac{b}{d}$ Then $a + r \cdot \sin(x + \phi) + c\cdot x\...
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If $\dim(V) $ is infinite, show that $V\oplus V$ is isomorphic to $V$

For a vector space $V$ of infinite dimension, to show that $V\oplus V$ is isomorphic to $V$ is to show that there exists an invertible linear transformation between $V \oplus V $ and $V$. Every ...
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9 views

Minimizing matrix norm via left-multiplication by $SL(m)$

Suppose that $M$ is an $m\times n$ matrix of full row rank, with $m \leq n$. Then if $\|M\|$ is the matrix norm induced on $M$ from the norm on our vector space, we can look for the following ...
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12 views

Find the all generalized eigenvectors of T

Let Define $T\in L(\mathbb C^2)\;by\\ T(w,z)=(z,0).$ Find the all generalized eigenvectors of T the standard basis of $C^2$ is {(1,0),(0,1),(i,0)(0,i)} T(1,0)=(1,0) then how can we write ...
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10 views

Find a volume, by parametrization sphere..

Consider a parametrization $C$ in $\mathbb R^3$ space, $x=\frac{4}{5}\cos t, y=1-\sin t, z=-\frac{3}{5}\cos t\;$ for some $t$. For the point $P$, moving on $C$, and the fixed point $Q$, moving on ...
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36 views

Color space as a vector space

I am not sure that this is the best place for this topic, so I apologize in advance. I have two questions. I think that color space with say additive colors (red, green, blue) forms a vector space. ...
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24 views

References for Duality Theory

I was wondering if anyone had any recommendations for Duality Theory. I've touched on Duality before in various courses but it's coming up quite a lot in my studies at the moment. I guess what I'm ...
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37 views

If $W$ is a subspace of $V$,then $\dim(W)+\dim(W^0)=\dim(V)$.

From the structure of this all i getting is that If $V$ an n-dimensional vector space with an ordered basis $\beta=(x_1,x_2,x_3,\dots,x_n)$ among them (say) first $k$-vectors form the basis ...
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1answer
37 views

Taking curl of Euler equation

Consider an inviscid incompressible flow. Euler’s equation can be written as $$\frac{\partial \textbf u}{\partial t} + \textbf ω × \textbf u = −\textbf∇\bigg( \frac pρ + \frac 12 \textbf u^2 + V \bigg)$...
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1answer
32 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 ...
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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 $r = (x, y, z)$...
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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 ...
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1answer
51 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
92 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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30 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 \gamma=...
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1answer
35 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), (2,3,0,5,-2)\...
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1answer
85 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
33 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
22 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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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 $...
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1answer
24 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 basis....
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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 ...
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0answers
27 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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25 views

Exist this vectorial equality, and this is correct?

doing a problem about distance with vectors appears this identity: $$\vec{A}\times\vec{B}=\frac{\vec{A}(\vec{A}\cdot\vec{B})-\vec{A}^{2}\vec{B}}{\lvert\vec{A}\lvert}=(\vec{A}\cdot\vec{B})\hat{A}-\...
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0answers
12 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) \hat{\...
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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 ...
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1answer
35 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 $R^4$?
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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 ...
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29 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 $V_i=Ke_1+Ke_2+\dots+Ke_n$...
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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 ...
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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
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1answer
30 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, \...
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2answers
21 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 ...
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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 \\ -2\end{pmatrix},\;\begin{...
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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 \...