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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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 $\mathbb{R}$ ...
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1answer
11 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: $$x=x_r+x_n$$ ...
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12 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 ...
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10 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 ...
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1answer
18 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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14 views

Subspace in R3 that has dimension 2 [on hold]

An example of a subspace in R^3 that has dimension 2 and contains the vector (1,0,1).
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23 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) = ...
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24 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 ...
3
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2answers
683 views

Sum of projections

Let $E_1$ and $E_2$ be projections on $V$, a vector space over $F$. Why is if $\operatorname{char}F\neq2$ then $E_1+E_2$ is a projection iff $E_1E_2=E_2E_1=0$ ?
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11 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?
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1answer
6 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 ...
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2answers
30 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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1answer
92 views

For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$. I think it is a subspace, but not 100% sure. I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$ ...
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3answers
165 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 ...
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0answers
12 views

Any technique calculate cut space without complement of graph?

I am studying sparse graphs, having a sparse adjacency matrix, where their complements tend to be dense (not sparse). The cut space for undirected graph is defined in terms of the complement. So a ...
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0answers
7 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, ...
0
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1answer
21 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
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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 ...
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1answer
14 views

Find the bases of the vector space of terminal sequences

Let V be the vector space of the sequences $ a = (a_0 , a_1 , a_2 , ...) $ of real numbers who are terminally - finally zero sequences (There is $ N $ such that $ a_n = 0 $ for every $ n > N $ ). ...
0
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1answer
14 views

Use the Gram-Schmidt procedure to construct orthonormal bases for the subspaces of Rn spanned by the following set of vectors

For part c: How can I quickly tell that the dimension of the subspace is 2? I used the algorithm and got "3" basis vectors before realising that the 3rd one was parallel to one of the others and ...
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1answer
14 views

signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
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1answer
24 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
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0answers
14 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
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0answers
23 views

bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
3
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0answers
94 views

Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
1
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1answer
30 views

If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
2
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1answer
139 views

log norm inequality for lower triangular part of matrix

Suppose $L$ is the lower triangular part of a matrix $A \in \mathbb{C}^{n\times n}$. Prove that $||L||_2 \leq ||A||_2 \log_2(2n)$. Here $||\cdot||_2$ is the matrix norm induced by the $p=2$ vector ...
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4answers
46 views

What is a simple means of proving that 3 vectors belonging to $\Bbb{R}^2$ are linearly dependent?

For my linear algebra class, there is a 2 part problem that asks, given the set {(1 2), (-1 -1), (1 0)}, prove or disprove that it is linearly independent using the definition only AND then prove or ...
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0answers
27 views

Show that $\mathcal{F}$ is a lattice fulfilling Stone's Axiom

Consider $$ f^+(0):=\lim_{r\searrow 0}\frac{f(r)}{r},~~~~~~\mathcal{F}:=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, f^+(0)\text{ exists}\right\}. $$ Moreover, let $\mathcal{F}^+$ be the set of all ...
2
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2answers
31 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
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0answers
29 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
14 views

Showing that $Im(L^*)=(Ker\: L)^\perp \space \:\mathrm and \:\:Ker(L^*)=(Im\: L)^\perp$

Let $V,W$ be finite-dimensional euclidean or unitarian Spaces and $L: V \to W$ a linear map. I have to show the following: $$Im(L^*)=(Ker\: L)^\perp \space \:\mathrm {and} \:\:Ker(L^*)=(Im\: L)^\perp ...
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1answer
24 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwarz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| = ...
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127 views

Absorbing sets on a vector space

The following definition for absorbing set is base in here. With this definition, is it true that finite intersection of absorbing sets is also absorbing?
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7answers
2k views

Why a subspace of a vector space is useful

I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, ...
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4answers
4k views

*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
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1answer
66 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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7answers
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Linear Algebra with functions

Basically my question is - How to check for linear independence between functions ?! Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$ Be a group of real valued fnctions. i.e ...
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1answer
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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5answers
936 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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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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0answers
27 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
67 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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1answer
24 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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2answers
20 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 + ...
8
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3answers
197 views

When is $V=U\oplus U^{\perp}$?

Let $V$ be a (infinite dimensional) vector space with inner product $(,)$ and $V$ may not be complete with the metric induced from the norm. Let $U$ be a subspace of $V$. What is the necessary and ...
2
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4answers
137 views

How to prove $ A^{\perp} $ is a closed linear subspace?

Suppose $ X $ is an inner product space and $ A\subseteq X $. I need to prove that $ A^{\perp} $ is a closed linear subspace of $ X $. Can anyone give me a idea?
1
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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 ...
0
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1answer
22 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? ...