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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15 views

Decompose $(\mathrm{Sym}^2 \mathbb{C}^2) \otimes (\mathrm{Sym}^2 \mathbb{C}^2)$ into irreducible representations of $\mathrm{SL}_2 \mathbb{C}$

Question: Let $V=\mathbb{C}^2$ be the standard representation of $\mathrm{SL}_{2}\mathbb{C}$. Decompose $(\mathrm{Sym}^2 V)\otimes (\mathrm{Sym}^2 V)$ into irreducible representations $\mathrm{SL}_2 ...
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1answer
46 views

Does a basis for $V \otimes_{\mathbb{F}} W$ always consist of pure tensors?

Given a field $\mathbb{F}$ and two $F$-vector spaces $V$ and $W$, it's true that if $\{v_i\}$ and $\{w_j\}$ are bases for $V$ and $W$, respectively, then the set $\{v_i \otimes w_j\}$ is a basis for ...
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1answer
26 views

$A$ be $n×n$ matrix $A^{n}=0$ ,$A^{n-1}$ not equal to zero a vector $v$ belongs to R^n.then how to proof {V,AV,…A^(n-1)V} is a basis. [duplicate]

Given $A$ be $n×n$ matrix such that $A^{n}=0$, but $A^{n-1}$ not equal to zero a vector $v$ belongs to $\Bbb{R}^{n}$. Proof that {$V,AV,\cdots,A^{(n-1)}V$} is a basis.
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1answer
49 views

Transpose of the differentiation operator

Please help me write down a step by step solution to the following problem Let $n$ be a positive integer and let $V$ be a finite dimensional vector space of all polynomial functions over the field ...
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1answer
30 views

Multiple parametric equations for planes and lines $\mathbb R^3$?

I want to know if you can get different sets of parametric equations for a particular line or plane in $\mathbb R^3$? The reason being I know you can have multiple directional vectors or normal ...
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1answer
24 views

Tangent Surface to a 4D Surface

I have been typing up notes for Multivariable Calculus. While doing so I have been pondering the terms I ought to use for higher dimensional surfaces and the associated tangent surfaces. With a curve ...
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3answers
129 views

Concerning $f(x_1, \dots , x_n)$

I am not getting even an intuition as how to do this problem. Please help me with a solution.. Let $n$ be a positive integer and $F$ a field. Let $W$ be the set of all vectors $(x_1, \dots , x_n)$ in ...
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1answer
55 views

Prove that the set of commuting matrices is a vector space

Prove that the set of real commuting matrices with the matrix $A= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ ...
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2answers
94 views

How to make sense of this linear algebra question about union of proper subspaces

I am having trouble understanding the following; I want to show that a vector space can never be written as the union of two proper subspaces, were proper subspace refers to being a subspace, yet not ...
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1answer
30 views

$f(\alpha _I) \ne 0$

I need help in this question... Let $F$ be a field of characteristic zero and let $V$ be a finite dimensional vector space over field $F$. If $\alpha _1,\dots , \alpha_m$ are finitely many vectors in ...
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3answers
50 views

Finding the closest point on a plane in $\mathbb R^3$ to the origin given its parametric equations?

I have a plane in $\mathbb R^3$ and I've found its parametric and plane equations. I thought of setting variables in the parametric equations to zero but, I keep getting different values for the ...
4
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1answer
37 views

Triangularization of matrices over algebraically closed field

A friend of mine is studying physics in first semester and for his next assignment, he has to prove the following theorem: Let V be a finite dimensional vector space over an algebraically closed ...
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2answers
35 views

Kernels and Epimorphisms (“Epic Morphisms”) as quotients of vector spaces

Let $f: A \to B$ be a monomorphism of vector spaces. We know that $\text{coker}(f)= B/A$. Is there a similar relationship between $\ker(g)$ and quotients with $M$ and $N$ given an epimorphism $g: M ...
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1answer
73 views

Can planes move like vectors?

I know that if I have a vector in $R^3$ I can move it around wherever I want, does the same apply to planes? Or are planes constrained to a location? The reason for this questions is that I was given ...
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0answers
9 views

Sum and intersection of annihilators. [duplicate]

I am a beginner in this course, please give me a detailed step by step solution of this problem. Let $W_1$ and $W_2$ be subspaces of a finite dimensional vector space $V$. If $X$ is a subspace of ...
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1answer
13 views

$g(\alpha )=f(\alpha )$

Can anyone help me prove the following? Let $V$ be a finite dimensional vector space over field $F$ and let $W$ be a subspace of $V$. If $f$ is a linear functional on $W$ , prove there is a linear ...
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1answer
18 views

Is U union W necessarily a vector space? [duplicate]

Let U and w be two subspaces of the same vector space V.Is U union W necessarily a vector space?
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0answers
13 views

Find all combinations of a vector in vector space to form a basis with existing vectors

Let $p_1(x)=1,p_2(x)=3x^2,p_3(x)=x+x^2-3x^3$ are given vectors from vector space $\mathbb{R}_3[x]$. Find all vectors $p_4(x)$ such that the set $\{p_1,p_2,p_3,p_4\}$ is a basis. Check if the found set ...
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0answers
11 views

Reference for the “geometry” or “arrangements” of subspaces of a vector space?

Inspired by Section $5$ of Chapter $1$ in Kostrikin & Manin's famous "Linear Algebra and Geometry", I am searching for a book or paper on the geometry or arrangement of subspaces in a ...
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1answer
39 views

Unique set of basis vectors

I am interested in finding conditions for a unique set of basis vectors in a finite dimensional vector space. Consider some finite dimensional vector space. Then there is an infinite number of sets ...
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0answers
17 views

subspaces of $\mathbb{C}^{2 \times 2}$ that are closed under multiplication

is there a technique to find out which subspaces of $\mathbb{C}^{2 \times 2}$ are closed under multiplication? The null space is of course always closed under multiplication, in the case of ...
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3answers
47 views

Proof that there exist constants $a_i$ such that $\int_0^1 f(x)e^xdx=\sum_{i=1}^na_if(i)$ for polynomial $f(x)$ of degree less than $n$

How do I show that for positive integer $n$ and $f(x)$ all real polynomial functions of degree less than $n$ there exist constants $a_i$ such that $$\int_0^1 f(x)e^xdx=\sum_{i=1}^na_if(i)?$$ I thought ...
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1answer
39 views

Drawing a perpendicular line from any angle

I'm trying to draw a line perpendicular to a line from any angle. I found following solution: http://math.stackexchange.com/a/1107295/303637 The slope between the given points is $$ m = \frac{3 - ...
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0answers
100 views

Given triangle ABC, how to move point B to a certain angle given that its new location lies within the direction of its old altitude.

I have a 2D coordinate system for 3 known points $A$, $B$, $C$. Given that I can only move point $B$, how can I compute for its new coordinate with a certain angle $\theta$ considering that its new ...
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0answers
10 views

Cartesian Decomposition.

I just read this on some notes written by my professor. It requires $X$ to be a linear map from complex Hilbert space $\mathcal{H}$ to itself, and that the Cartesian decomposition of $X$ is $X = H + ...
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1answer
36 views

How to calculate sum of vector subspaces

How do you sum these given subspaces? $$S_1=\{(x,y) \in R^2 | x=y\}$$$$S_2=\{(x,y) \in R^2 | x=-y\}$$ The book that I am currently learning from gives the answer to be $R^2$, but how do you get there? ...
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1answer
37 views

Finding the closest point on a plane and a given point

I have a plane, and a point and I am able to interchange between the plane equation and parametric equation quite well now. I recently figured out how to find a point on the line thats the closest ...
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2answers
26 views

Given a lines parametric equations, and a point how do I find the closest point on that line to that point.

I thought of using the dot product set to $0$ but I'd need two vectors, and I online have one if I use the parametric equations as $x, y, z$ values of a vector. This is the example Line: $l = ...
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2answers
46 views

Prove that $\mathrm{span}(S) = S$ for a subspace $S$.

Prove that if $S$ is a subspace of a vector space $V$, then $\mathrm{span}(S) = S$. What I tried: I considered using the properties of vector spaces or maybe using an example where $S \subseteq ...
3
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1answer
44 views

What is the relationship between a quotient space and annihalator?

If we have a vector space $V$ and subspace $W$, we have that $$\dim(V/W) = \dim V - \dim W.$$ Similarly for the annihilator $W^{\circ}$ we have that $$\dim W^{\circ} = \dim V - \dim W.$$ What is ...
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0answers
33 views

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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1answer
33 views

Invariant subspace under nilpotent operator

Let $V$ be a vector space, dim$V=n$, $N\in L(V)$ a nilpotent operator of index $n$. Let $W$ be an $N$-invariant subspace of $V$ and $m$ the nilpotency index of $\left.N\right|_W$. Prove that ...
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1answer
35 views

Prove that the set of vectors is a subspace

Let $B=\{b_1,b_2,b_3\}$ and $C=\{c_1,c_2,c_3\}$ are two basis of a vector space $W$ over $\mathbb{R}$ and $b_1=2c_1−c_2−c_3,b_2=−c_2,b_3=2c_2+c_3$. Prove that the set of all vectors which have the ...
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0answers
29 views

2 forms and Base

Let$\: V \;$ be a n-dimensional vector space and $\:w\;$ a two form. Proof that there exists a base $\alpha_1,\alpha_2,..\alpha_n, \in V^* \;$ so that $\; \omega =\alpha_1 \wedge \alpha_2 + \alpha_2 ...
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1answer
8 views

Bounds on the sum of the elements of unit-length complex vector

Given an $n$-element complex vector $\mathbf{x}=[x_1,\ldots,x_n]\in\mathbb{C}^n$, where $\|\mathbf{x}\|_2^2=\sum_{i=1}^n|x_i|^2=1$, I am wondering if anything can be said about the product $A\bar{A}$ ...
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0answers
22 views

Linear mapping of lines passing through origin vector in $\mathbb{R}^n$ to $\mathbb{R}^m$

I've this question. Prove that if $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear map, then $f$ must map lines passing through the origin vector in $\mathbb{R}^n$ to either lines passing through ...
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1answer
35 views

Vector valued measure concept from Rudin (Functional Analysis).

I've a question, from rudyn functional analysis, by studying the spectral theorem i've seen the concept of "vector valued measures", i haven't seen in the book an explanation of what that means and ...
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1answer
41 views

A property about normal operator

Let $H$ be a Hilbert space, $L$ be a normal operator (i.e.$LL^*=L^*L$). Prove that there exist a unitary operator $U$ such that $L^*=UL$
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1answer
36 views

How to extend a unitary operator to a larger space?

Suppose $V$ is a Hilbert space with a subspace $W$. Suppose $U: W\rightarrow V$ is a linear operator which preserves inner products. Prove that there exists a unitary operator $U':V\rightarrow V$ ...
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4answers
410 views

Why do these vectors not belong to the same vector space?

I'm trying to verify that $W$ (being the set of all vectors in $\mathbb R^3$ whose third component is $-1$) is not a subspace of the vector space. You can have a vector $(0,0,-1)$ and through a ...
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1answer
18 views

Polarization Identity for Complex Scalars

So I was trying to prove that for $x,y\in \mathbb{C}$ we have that: $4 \langle x,y \rangle=||x+y||^2-||x-y||^2+i||x+iy||^2-i||x-iy||^2$. I got that $||x+y||^2-||x-y||^2=4\Re\langle x,y \rangle$ and ...
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1answer
43 views

Proof of $dim\ ker\ (f \circ g) \le dim\ ker\ (f) + dim\ ker\ (g)$

In my lecture notes I have an inequality for any linear transformations $f: X \to X$, $g: X \to X$ for any finite-dimensional vector space $X$: $$dim\ ker\ (f \circ g) \le dim\ ker\ (f) + dim\ ker\ ...
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1answer
21 views

What is the probability of choosing a vector that is not a linear combination of $k$ independent vectors in $\mathbb{B}^n$?

What is the probability of choosing a vector that is not a linear combination of $k$ independent vectors in $\mathbb{B}^n$? My guess is that it can have $n - k - 1$ of its elements as being $0$, so ...
4
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1answer
41 views

Find a basis for the space of quotients of polynomials over $\mathbb{R}$

$F$ is a field and define $F(t):=\lbrace \frac{p}{q} \mid p,q \in F[t], q\neq0 \rbrace$. Find a basis for $\mathbb{R}(t)$. I know that the basis for the polynomials over $\mathbb{R}$ is $\lbrace ...
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0answers
32 views

Is the map $F:(\mathbb{R}^2)^{\mathbb{R}} \rightarrow \mathbb{R^2}, \phi \mapsto \phi(1)$ linear?

I am supposed to prove or refute that $F:(\mathbb{R}^2)^{\mathbb{R}}\rightarrow \mathbb{R^2}, \phi \mapsto \phi(1)$ is a linear map, with $(\mathbb{R}^2)^{\mathbb{R}}$ being the set of all functions ...
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1answer
45 views

Are some of the axioms of a norm of a vector space unnecessary?

I have a homework problem where my task is to find out if some of the axioms of a norm of a vector space are unnecessary, meaning they can be derived from other axioms (I presume from the problem ...
1
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2answers
24 views

in a linearly dependent set there exist a vector which is a linear combination of finite number of other vectors

I'm watching Linear Independence and Subspaces lecture, and the prof proves that in any linear dependent space there's a vector which is a linear combination of finite number of other vectors. He ...
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2answers
33 views

How to find a basis for this $\Bbb R$-vector space : $\{(x, y) \in \Bbb C^2 \mid x + iy = 0\}$?

I tried this : $\{(x, y) \in \Bbb C^2 \mid x + iy = 0\} \iff \{(x, y) \in \Bbb C^2 \mid x = -iy\} \iff \{(-iy, y) \in \Bbb C^2\} \iff \{y(-i, 1) \in \Bbb C^2\}$ So a basis for that vector space ...
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0answers
30 views

How to invert a transformation

I've come across a recursive equation involving vectors. You basically have one starting point $P = (x, y)$ and you transform it to another point $P'=(x', y')$ with the following equations $$ x' = x ...
2
votes
2answers
22 views

Circular reasoning in a simple consequence of vector space definition?

In books, after the definition of vector spaces, one usually proves simple consequences of it, such as $(-1)\cdot v=-v$. One of these consequences is the following: If $\alpha\cdot v=0$, then ...