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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8 views

3D point rotation round a fix reference point

I want to compute a transformation from 3D point A to 3D point B through a reference point 0 which is fixed. I have the 6DOF transformation from A - 0 and B - 0. That is x,y,z and Quaternions of ...
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18 views

The subset of a set of vectors such the this subset has the points at most extreme values for all dimentions

Condiser $X\subset \mathbb{R}^n$ We can define some the set of per axis outermost points, on a per axis: $$Out(X) = \{ x \mid x \in X \: \wedge \: (\exists i\in[1,n], x_i=\underset{y\in ...
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2answers
21 views

Uniqueness of isomorphism to linear spaces.

Even if an isomorphism between two linear spaces $L$ and $M$ over a field $\mathbb{K}$ exists, it is defined uniquely only in two cases: $L=M=\{0\}$ and $L$ and $M$ are one-dimensional, ...
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1answer
43 views

Generating $\mathbb R^4$

Assume that we have $6$ vectors in $\mathbb R^4$ such that every two of them is independent. can we generate $\mathbb R^4$ with them?
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33 views

Is the intersection of an infinite family of subspaces of $V$ itself a subspace of $V$?

Given $\{U_i\}_{i\in\mathbb N}=\{U_1,U_2,U_3,...\}$ an infinite family of subspaces of $V$ is $\bigcap_{i\in\mathbb N}U_i$ a subspace of V? I know that it's right for $n$ subspaces with a pretty ...
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1answer
50 views

Find eigenvalues of operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$. Suppose we know its eigenvalues - $\lambda_1, \lambda_2, \ldots, \lambda_n.$ Now consider the ...
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2answers
96 views

Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?

Can we define a vector space structure on $\mathbb {R}^n$ other than usual scalar multiplication and usual addition such that the dimension of $\mathbb {R}^n$ over $\mathbb {R}$ is not $n$ but some ...
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22 views

This vector space isomorphism $\phi$ $\mapsto $ $\phi^{*}$ between $V$ and $V^{*}$ is not a ring isomorphism

V is a vector space of finite dimension $n$ over a field $F$. $V^{*}$ is the dual space of $V$ . For $\phi$$\in End(V)$ we define $\phi^{*}$$\in$$End(V^{*})$ as $\phi (f) = f\circ ...
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1answer
21 views

Equation of the plane tangent to the given surface

Find the equation of a plane tangent to the surface given by $$xyz+x^2-3y^2+z^3=14$$ at $$P=\left( 5,-2,3 \right)$$ In my opinion answer is: $$4x+27y+25z-41=0$$ If not please tell me what am i doing ...
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2answers
37 views

What should we understand from the definition of orthogonality in inner product spaces other than $\mathbb R^n$?

In the beginning of linear algebra courses, there are vectors in $\mathbb R^n$ and the dot product is introduced. We learn that if the dot product of two vectors is zero, then these vectors are called ...
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16 views

Representing ordered sequence as a vector

What is the best way to represent a bunch of ordered sequence as a vector in a d-dimensional space? Imagine we have some ordered sequences like this: ...
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2answers
18 views

Orthogonal unit vectors to 2 vectors

My question is: Find 2 unit vectors orthogonal to both vectors: (1,-1,1) and (0,4,4) So far, what I have done is constructing a line between the two vectors: (1,5,3). What should i do next?
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1answer
29 views

Equation of the plane tangent to the given area

Find the equation of the plane tangent to the surface: $$x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}=1$$ at the point: $$P=\left(1,-1,1\right)$$ How to find it? I know i have to calculate a ...
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1answer
32 views

Finding a perpendicular vector

This is given: $P = (-1,1,0),\; Q = (1,5,6)\; \text{and} \; R = (3,-1,4).$ My question is: Find the values of $x$ (where x is a real number) for which $PR + xQR\;$ is perpendicular to $PR$. So far, ...
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0answers
46 views

Countably many projections on more than continuos vector space with trivial commutant?

Is there such an example? An $\mathbb{F}_2$-vector space $V$ of dimension strictly more than the continuos $c=|2^{\mathbb{N}}|$, and a numerable set of commuting $\mathbb{F}_2$ projections ...
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1answer
36 views

Proof: Set of vectors is not a subspace.

I am working through my textbook and am stuck on this example problem: Prove that the set of vectors $S = \{(x_1,x_2,\dots,x_n) \mid x_1 + x_2 + \dots + x_{n-1} \geq x_n \}$ in $\mathbb R^n$ for $n ...
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0answers
11 views

Rotating a RotationTranslation Matrix?

Good day, I have been given a $4\times 4$ homogenous $\text{RT}$ Matrix, that maps frame $C$ to $A$. I am then tasked to rotate that frame $C$ around the $Z$ axis w.r.t. $B$, to give a new frame ...
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1answer
22 views

Omission in Jacobson's BAI regarding extension of isometries.

Suppose $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ equipped with a nondegenerate quadratic form $Q$. Witt's cancellation theorem says that if $U_1,U_2$ are ...
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3answers
34 views

Is this linearly independent? What is the dimension span?

Consider the vectors in $\mathbb{R}^4$ defined by $v_1 = (1,2,10,5), v_2 = (0,1,1,1), v_3 = (1,4,12,7)$. Find the reduced row echelon form of the matrix which has these as its rows. What is its ...
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1answer
56 views

Let $x,y \in \mathbb C^n$ , where $n>1$ ; then does there exist symmetric $A \in M_n(\mathbb C) $ such that $Ax=y$?

Let $x,y \in \mathbb C^n$ , where $n>1$ ; then does there exist $A \in M_n(\mathbb C) $ such that $Ax=y$ ? Can we find such a symmetric matrix $A$ ?
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0answers
32 views

What are complex and real dimensions of this space?

Show that the vectors $v_1$ = (i,1+i,2+i), $v_2$ = (1,1+i,2+i), and $v_3$ = (2,-i,-i) form a basis for the complex vector space $C^3$.... Show that $v_1,iv_1,v_2,iv_2,v_3,iv_3$ is a basis of ...
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1answer
36 views

Relationship between isomorphic vector spaces and inner product

Let $V$ and $W$ be isomorphic vector spaces. "If $\langle\vec u, \vec v\rangle$ is an inner product in $V$, then it is also an inner product in $W$". Does such a relationship exist?
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43 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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1answer
38 views

What do double parallel lines on vectors mean? [closed]

What do the lines mean in the notation $\|u\|$ where $u$ is a vector?
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1answer
53 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
2
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1answer
30 views

Basis for a linear space

Let the space of polynomials of $x$ of degree $\leq n-1$ with coefficients in the field $\mathbb{K}$. How can I show that $\{1, x-a, (x-a)^2, \cdots, (x-a)^{n-1}\}$ form a basis for this linear ...
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1answer
15 views

Orthogonal complement of a subspace

I don't understand how to visualize these statements algebraically (are they true or false??): Let $S$ and $T$ be subspaces of $E$: I) $(S+T)^\perp \subset S^\perp \cap T^\perp$ II) $S^\perp + ...
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2answers
40 views

Showing 2 vector spaces are isomorphic.

I am trying to understand how to show two vector spaces are isomorphic. You do this by showing there is an isomorphism that can be mapped between the two spaces. What I don't understand is my ...
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1answer
29 views

Is frequency a scalar quantity?

Well, our professor in class posed a question to all of us: Is frequency a scalar quantity? The obvious answer, of course, is yes, it is, because it does not have a direction associated with it. But ...
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0answers
48 views

Let G be an abelian group, and V be a faithful irreducible representation of G over C

This seems kind of obvious to me but I'm really having trouble thinking of what to do! Any help would be appreciated. Let G be a finite abelian group, and V be a faithful irreducible representation ...
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3answers
39 views

Vector spaces $V=-V$

I have a question about vector spaces: Since vector spaces have additive inverses, does that mean that $-V=V$?
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2answers
307 views

Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...
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1answer
16 views

If $\lim\limits_{x\rightarrow 0}(f(x)\cdot y)=0\space\forall y\in\mathbb{R}^n$, show that $\lim\limits_{x\rightarrow 0}f(x)=0$

Let $f(x):\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function such that $$\lim\limits_{x\rightarrow 0}(f(x)\cdot y)=0\space\forall y\in\mathbb{R}^n$$ Show that $\lim\limits_{x\rightarrow ...
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1answer
40 views

Dual Space Isomorphism

If $V$ is a finite dimensional real vector space. Let $$ V^* = \{f: V \to \mathbb{R} : f ~\text{is linear}\} $$ (Note $V^*$ is called the dual space of $V$.) Prove the vector spaces $V$ and $V^*$ ...
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2answers
62 views

V is a linear space that has finite dimension $T : V \rightarrow V$ Proof if $Ker T = Ker T^2$ then $Im T = Im T^2$

I have the following question : V is a linear space that has finite dimension $T : V \rightarrow V$ Proof if $Ker T = Ker T^2$ then $Im T = Im T^2$ I tried to proof it, but I got stuck, This is what ...
2
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1answer
34 views

Any tetrahedral geometry theorems of methane bonding angles?

For my 12 grade folio task I need to find alternate ways of finding the bonding angles in a methane molecule (regular tetrahedron). I have already done it through vector methods, co-ordinate geometry ...
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1answer
17 views

volume of the parallepiped spanned by the vectors

Hi I am having difficulty with part (2) of the following proposition. Suppose that $x,y,z\in\mathbb{R}^3$, then (1) $\|x\times y\|=\|x\|\|y\|\sin\theta$ is the area of the parallelogram spanned by ...
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0answers
29 views

Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
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1answer
36 views

Sums of vector space and dimension

In many questions, I see that we have that given $X,Y$ as subspaces of $R^n$ then $V=\{ x+y \mid x \in X, y \in Y\}$ is a subspace. I understand the proof of this. I am not sure why the subtlety that ...
5
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1answer
36 views

Closed formulas for two Poincaré series

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative ...
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2answers
79 views

Exists polynomial satisfying following?

Let $s, u \in M_m(\mathbb{k})$ be a pair of commuting matrices such that $s$ is a diagonal matrix and $u$ is a strictly triangular matrix (with zeros on the diagonal). Put $a = s + u$. Does there ...
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1answer
47 views

What space is the set of all CDFs?

I'm relatively new to functional analysis and am trying to make comparisons across different CDFs (cumulative density functions), i.e. right-continuous, weakly increasing functions ...
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4answers
73 views

Not understanding the case where $G$ is abelian with every element of order $2$

Suppose an abelian finite group $G$ (with $o(G)>2$) has every non-identity element of order $2$. Show that there exists a non-trivial automorphism on $G$. After a bit of searching, I found a ...
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1answer
54 views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
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2answers
29 views

Linear algebra, sub spaces, dimension, span

Consider the subspaces $A=span\{(1,-1,0,0),(0,0,1,-1)\}$ and $B=span\{(1,0,-1,0),(0,1,0,-1)\}$ of $\mathbb{F}^4$. A) what are the dimensions of these subspaces? The dimension of both subspaces is ...
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2answers
19 views

Determine the set is a subspace of vector space

Determine whether the following sets $U$ is a subspace of the indicated vector space $V$. Justify your answer. $$V =\mathbb{R}^3 , U =\{(a,b,c)|a,c \in\mathbb{R}\}.$$ How do we determine and ...
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0answers
23 views

Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star ...
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2answers
53 views

The kernel of an element of End(End(V)) for V a finite dimensional vector space

I am a grad student currently studying for an upcoming algebra qualifying exam. I have been working through previous exams and I have gotten through most of them, but I am stuck on this question: ...
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1answer
31 views

Vector Space Spanned by Legendre Polynomials

Problem Let $V$ be the vector space over $\mathbb{R}$ spanned by Legendre Polynomials: ...
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1answer
23 views

Vector norm - Understanding the definition of the unit sphere

If $\|x\|=1$ just means the vector $x$ has length one - Then why is the unit sphere defined as $S=\{x\in X| \quad \|x\|=1\}$? let $X$ be a normed linear space with the Euclidean norm, then letting ...