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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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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10 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

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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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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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
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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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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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

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 ...
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2answers
964 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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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
34 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
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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1answer
40 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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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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5answers
147 views

What is the rule for using $| \cdot |$ and $\| \cdot \|$ in Cauchy-Schwarz inequality

In this widely cited and wildly popular proof of the Cauchy-Schwarz inequality, the authors write (http://www.math.lsa.umich.edu/~speyer/417/CauchySchwartz.pdf) Let $u$ and $v$ be two vectors in ...
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1answer
34 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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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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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
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 ...
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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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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
377 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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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 ...
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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 ...
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2answers
21 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 ...
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1answer
22 views

If $\{v_1, v_2, …, v_n\}$ is a basis and $f$ is an injective morphism, show that $\{f(v_1), f(v_2), …, f(v_n)\}$ is linearly independent.

Let $V_1$, $V_2$ be two $K$-vector spaces with $dim_K V_1 = dim_K V_2 = n$, $f:V_1 \rightarrow V_2$ a morphism and $B = \{v_1, v_2, ..., v_n\}$ a basis for $V_1$. Now consider the set $T = \{f(v_1), ...
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2answers
378 views

To prove any two basis of Finite Dimensional Vector Space have same number of elements

To prove any two basis of Finite Dimensional Vector Space have same number of elements If i take bases as $S_!$ = {$\alpha_!$ ,$\alpha_2$ ,....$\alpha_n$ } $S_2$ = {$\beta_!$,$\beta_2$ .... ...
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2answers
4k views

Vector line parallel to $x$-axis?

The points $P$ and $Q$ have position vectors, relative to the origin $O$, given by $$ \overrightarrow{OP} = 7\mathbf{i} + 7\mathbf{j} - 5\mathbf{k} \quad\text{and}\quad \overrightarrow{OQ} = ...
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1answer
14 views

A problem involving proving cyclic module defined via invertible linear operator has cyclic inverse module

I have met this recently in my abstract algebra course dealing with modules over PIDs and we are dealing with cyclic modules at the moment, the problem I am having difficulty with is as follows: ...
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3answers
149 views

Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
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0answers
34 views

Is there a relation name that only applies to equal vectors if they also have the same origin?

Vectors $\vec{A}$ and $\vec{B}$ are equal if they have the same magnitude and direction. Is there a mathematical relation defined to apply to them only if they also have the same origin? Actually, I ...
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4answers
22 views

Find out if point is touching line

So, let's assume I have a line that made up of point one (x1, y1) and point two (x2, y2). Then I have a third point somewhere in two dimensional space (x3, y3). I would like to find out if this point ...
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1answer
54 views

Find dimension and a basis of a subspaces $U+V$, $U \cap V$ in terms of the parameter $\alpha$

Let $U=span((1,1,1,2),(1,2,2,\alpha))$ and $V=span((1,3,4,\alpha+2),(1,4,\alpha,\alpha+1))$ are the subspaces of $\mathbb{R^4}$. ($\alpha\in\mathbb{R}$). Find dimension and one basis of $U+V$, $U ...
3
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1answer
31 views

Basis of a subspace with a moving vector

I have a problem and I don't know if there is a solution to it. I try to explain you an example which is simplified with respect to the scenario I have but it gives you a good idea of what my ...
2
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2answers
447 views

Space spanned by matrices

I have a set of $5$ by $5 $matrices, $M_1,M_2,...,M_{19} ,M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should ...
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1answer
35 views

Do eigenvectors with pairwise distinct eigenvalues of a bounded, linear, nonnegative, symmetric operator on a Hilbert space build an orthogonal basis?

Let $H$ be a Hilbert space and $Q$ be a bounded, linear, nonnegative and symmetric operator on $H$ with finite trace. By the Hilbert–Schmidt theorem, there is an orthonormal basis ...
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1answer
25 views

Show permutation representation is reducible, by finding G-invariant subspace [duplicate]

$(\pi,V) $ is the permutation representation of the symmetric group $S_5 $, $ V=C^5$ and the action of standard basis vectors of $ V$ is given by $\pi(\sigma)e_i=e_{\sigma(i)} $ for $\sigma\in S_5 $ ...
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3answers
298 views

Cannot understand how angle between two vectors is calculated

On the picture below I am not getting why we calculate $\cos^{-1}(\frac{1}{3})$ instead of $\cos(\frac{1}{3})$. Sorry if the question is dumb.
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1answer
532 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
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2answers
29 views

Is it true that any set of functions that have a finite codomain are not vector spaces over R?

I am asking this question to clarify my understanding of set of functions as vector spaces. For instance, if i have the function $$f:[a,b]->[c,d]$$ Where a,b,c,d ∈ R Then $$f(s)∈[c,d]$$ For any ...
2
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1answer
34 views

If $A_{n\times n}$ is nilpotent matrix then $\beta = (y, Ay,A^2y, \ldots, A^{k-1}y)$ is the basis of vector space $\mathbb R^n$. [duplicate]

$A_{n\times n}$ is nilpotent matrix, $A^k=0$ and $A^{k-1} \neq 0$. If $A_{nxn}$ is nilpotent matrix and $A^{k-1} \neq 0$ for some $y \in \mathbb R^n$, then prove that $\beta = (y, Ay,A^2y, \ldots, ...
2
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2answers
48 views

Are projections with the same kernels the same?

This question is related to another question I just asked. I thought I figured it out but I got confused again. Given two projections $k^n\rightarrow k^n$, represented by $n\times n$ matrices $A$ ...