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
19 views

What are those variations of norms called?

Let $V$ be a vector space with a function $\|\cdot\|$ on it that satisfies all the axioms of norms except for scalability condition $\|\alpha \mathbf{x}\| = |\alpha| \|\mathbf{x}\|$ replaced with ...
2
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1answer
18 views

Characterization of vectors via $\ell_p$ norms

Suppose you are given all $\ell_p$ norms of a vector $v\in \mathbb R^d$. Is it true that the set of all its $\ell_p$ norms $\{\|v\|_{p},p=1,..,\infty\}$ uniquely define the vector $v$ up to ...
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2answers
47 views

Proof for $V \cong V^{**}$

Theorem: Let $V$ be an vector space. Then the dual space of $V$'s dual space is canonically isomorphic to $V$. I am able to prove that $V$ is a subspace of $V^{**}$, the map ...
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1answer
17 views

Computing the characteristic polynomial

Consider the following matrix A over the field $F_7$ $$ \left(\begin{array}{rrr} 3 & 4 & 4 \\ 2 & 5 & 2 \\ 1 & 2 & 5 \end{array}\right) . $$ I'm asked to ...
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2answers
32 views

Show that the full null space of the matrix A and its column space in the plane 2x+2y - z = 0

Show that the full null space of the matrix A = $\begin{bmatrix} 0&1&5\\ 1&0&0 \\ 2&2&10 \end{bmatrix}$ is the line $\lambda$(0.-5,1), $\lambda \in \mathbb R^3$ and its ...
3
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0answers
29 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
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1answer
25 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
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2answers
62 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
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2answers
25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
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1answer
13 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
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0answers
25 views
+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
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0answers
12 views

Clarification on Sequence space

I have a trivial question but which I'm feeling confused. Is the sequence space a finite collection of vectors whose components are infinite or am I misunderstanding the concept?
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1answer
29 views

Endpoints of a 3D line

How to find the coordinate of the endpoints (A and B) of a line on a surface with known surface normal, center coordinate, and length?
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2answers
58 views

Linear Algebra: which of the definition of subspace of a vector space is more correct?

In a test I was asked to give a definition to a subspace to a vector space, I wrote: A subset $V$ is a subspace of $X$ if $0 \in V$ and $\forall u,v \in V, > \exists \thinspace V$ s.t. $ u+v = ...
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2answers
42 views

Vector space sizes

$T:V \rightarrow V$ where $V$ is a finite dimensional real vector space I can show that $\ker(T) \subseteq \ker(T^2) \subseteq \ker(T^3) \subseteq\cdots $ Prove there exists some $k$ such that ...
1
vote
1answer
20 views

Are scalar/vector fields in multivariable calculus related to fields of vector spaces in linear algebra

In linear algebra, I have learned that vector spaces are defined over fields. I have to admit that I don't have any background in abstract algebra, so my knowledge of fields are limited to $\mathbb R, ...
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1answer
42 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
0
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1answer
38 views

A problem from Finite Dimensional vector spaces

Problem : If $ M $ and $N$ are two subspaces of the vector space $V$ such that $\forall v \in V $ , $ v \in M $ or $ v \in $ (or both) . Prove that at least one of the is equal to $ V $ My ...
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0answers
14 views

What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
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2answers
56 views

A map that's 1-1 but not onto

I've got some confusion about the definition of a 1-1 map. When I searched for "1-1 correspondence" on Wikipedia, I got redirected to the "bijection" page. So I think the two words mean just the ...
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2answers
25 views

Use the cross product to find a parallel vector

I'm confused by this exercise here : Using the cross product, for which value(s) of t the vectors w(1,t,-2) and r(-3,1,6) will be parallel I know that if I use the cross product of two vectors, I ...
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votes
1answer
10 views

A plane and the matter of vector crossing order

I have three 3D points $A$, $B$ and $C$ which are defining a plane. If I want to get the equation of the plane, firstly I need its normal vector. Is it matter if I do it with $AB \times AC$ or $AC ...
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0answers
56 views
+50

When a vector space will be a complete lattice?

Let $E$ a vector space, and let $P$ a strict cone in $E$ (i.e) $P\subset E$ verify: $$ \mathbb{R}^+ P\subset P \\ P+P\subset P\\ P\cap (-P)=\{0\} $$ So we can easily construct a partial order on $E$ ...
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0answers
36 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
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0answers
12 views

Inner Product of Square Matrices

Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{n1} & \cdots ...
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0answers
11 views

Dot Product of Square Matrices & Inner Product

I need some help! Thank you in advance. Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots ...
1
vote
1answer
19 views

Proof that the set of integrable real-valued functions is a vector space

From Folland's Real Analaysis: Modern Techniques and Applications: Proposition: Let $(X,\mathcal{M},\mu)$ be a fixed measure space. The set of integrable real-valued functions on $X$ is a real ...
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0answers
13 views

Difference between orthogonal projection onto a point and onto a vector.

A trivial question although I'd like some good answers. Are there any mathematical difference? My vector calculus is a bit rusty.
0
votes
1answer
22 views

Finding the Jordan Normal Form for a General Linear Transformation

Hey everyone here's the problem: Let V be a vector space with dim(V)=n For a particular linear transformation,f, we are given that there are two distinct eigenvalues, λ1 and λ2, with corresponding ...
2
votes
1answer
64 views

Can I assume that the dimension of a vector space is always non-negative?

I'm trying to prove that if $V$ is finite-dimensional and $U_1,...,U_m$ are subspaces of $V$, then $\dim(U_1+...+U_m)\le \dim U_1+...+\dim U_m$ through induction. For $m=1$, the inequality is trivial ...
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votes
2answers
90 views

Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?

Is easy to show that $(\mathbb{R},\mathbb{Q},+,\cdot)$ is isomorphic to $(\mathbb{R}^n,\mathbb{Q},+,\cdot)$ as vectorial spaces and then $(\mathbb{R},+)$ isomorphic to $(\mathbb{R}^n,+)$. This result ...
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0answers
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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0answers
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
42 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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2answers
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 ...
1
vote
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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votes
2answers
93 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 ...
0
votes
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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0answers
15 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: ...
0
votes
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?
0
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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 ...
1
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1answer
31 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
35 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 ...
2
votes
1answer
20 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
32 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 ...
3
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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 $A$ ?