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
11 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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0answers
25 views

Showing that the characteristic equation is satisfied for a specific endomorphism

Let $V$ be an $n$-dimensional $\mathbb{R}$-space and $\alpha$ and endomorphism of $V$. I am trying to show that if $\{\mathbf{v},\alpha(\mathbf{v}),\dots ,\alpha^{n-1}(\mathbf{v})\}$ spans $V$ then ...
2
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1answer
33 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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2answers
22 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 ...
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0answers
17 views

Injective map from image $\to$ image of dual

Let $\alpha:\;V\to W$ be a linear map, with $V,W$ finite dimensional $\mathbb{R}$-spaces, where $W$ is equipped with an inner product. I am doing a problem which involves showing that there is an ...
3
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1answer
25 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
57 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 \\ ...
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0answers
8 views

How to define this space? (matrix of coordinates)

We will let $F$ denote an arbitrary field such as the real numbers $R$ or the complex numbers $C$. For any positive integer $n$, the space of all $n$-tuples of elements of $F$ forms an ...
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2answers
16 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
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2answers
16 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
27 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 $K_n[X]$?
4
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1answer
50 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 ...
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1answer
13 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
39 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 ...
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2answers
17 views

Line Integral Problem! Help! [on hold]

In $(x,y,z)$ space, we are given the following vector field, $$V(x,y,z)=(\cos(y), -x\sin(y) + \sin(z), y\cos(z) + 2)$$ and the points $$O(0,0,0), P(x_0, y_0, z_0), Q(1,0,π)\ \text{and}\ ...
0
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1answer
15 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 ...
1
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2answers
29 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 ...
1
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2answers
32 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 ...
38
votes
12answers
68k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
1
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1answer
31 views

Decomposing vector space into positive/negative definite subspaces

Consider the quadratic form: $$f:\mathbb{R}^3\to\mathbb{R};\quad (x,y,z)\mapsto x^2+2y^2-2xy-2xz$$ I am doing a problem which asks me to find subspaces $A,B\subseteq \mathbb{R}^3$ such that ...
7
votes
2answers
2k views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
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5answers
448 views

What does vector mean in Linear Algebra?

I have just started reading Linear Algebra and there are some basic things I cannot understand. I read some answers on this site and also tried to search in some books but I didn't find a clear ...
0
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1answer
33 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 $\mathbb{R}$ ...
0
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1answer
25 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
1answer
26 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 ...
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votes
0answers
23 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?
2
votes
3answers
276 views

Conditions - Linear Combination of 3 Vectors inside a Triangle (Strang P10, 1.1.20)

Under what restrictions on $c, d, e,$ will the combinations $c\mathbf{u} + d\mathbf{v} + e\mathbf{w}$ fill in [ie bridle/rein in] the dashed triangle? To stay in the triangle, one requirement is $c ...
1
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2answers
22 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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0answers
14 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
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1answer
21 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
20 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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0answers
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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0answers
27 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 ...
4
votes
2answers
692 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 $E_1E_2=E_2E_1=0$ ?
0
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1answer
12 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 ...
0
votes
2answers
34 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 ...
0
votes
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 $p_1(15)=0$ ...
4
votes
3answers
173 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
9 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
23 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 ...
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 ...
0
votes
1answer
15 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
votes
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 ...
0
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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 ...
0
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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 ...
2
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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, ..., ...
0
votes
0answers
24 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
votes
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
vote
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
votes
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 ...