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 a basis for the vector space of symmetric matrices with an order of $n \times n$ [duplicate]

Find a basis for the vector space of symmetric matrices with an order of $n \times n$ This is my thought: by definition of symmetry, $a_{i,j}=a_{j,i}$. Therefore, the basis should consist ${n^2-n} ...
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1answer
32 views

Dimension of ${U\cap W}$

I have the following question: Let $V$ be a vector spaces with dimension $n$. Let $U$ and $W$ be distinct sub vector spaces of $V$ with dimension $n-1$. Find the dimension of ${U\cap W}$. I proved ...
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2answers
98 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
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Notation question in Majda and Bertozzi's “Vorticity and Incompressible Flow”

On pg 2, the fluid velocity in the Navier-Stokes system of equations is noted as: $v(x,t) \equiv (v^1, v^2, \ldots, v^N)^t$, where I am assuming that the velocity vector field is time-dependent. The ...
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4answers
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Proof there is always a base in a v.s such that the coordinates of a vector are the elements of a given set

Given any non-null vector of a vector space over a field $K$, of finite dimension $n$, and given any ordered set of $n$ elements (not all null), all in $K$, prove that there exists a base such ...
2
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1answer
45 views

multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
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0answers
48 views

Decomposition of Vector into n vectors (with length conditions)

If i have a vector V and i want to decompose it to n vectors their lengths are $L_1, L_2, L_3, .. L_n$ Where $\ \vec{V}= ...
2
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0answers
110 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
2
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1answer
125 views

Subspace with different vector space operations

Let $A,B$ be vector spaces such that $A\subseteq B$. Is it true that $A$ is a subspace of $B$? I claim that the answer is no, because it is possible that $A$ and $B$ might be equipped with different ...
2
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1answer
27 views

Clarification regarding notation: a $\mathbb{C}[X]$ -basis of $\operatorname{Der}_{\mathbb{C}}\mathbb{C}[X]$

I came across the following sentence in an article: The derivations $\frac{\partial}{\partial F_1},\ldots,\frac{\partial}{\partial F_n}$ form a $\mathbb{C}[X]$-basis of ...
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2answers
418 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...
2
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2answers
273 views

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k<n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has ...
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1answer
948 views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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1answer
54 views

Give an example of a linear transformation

$$\begin{array}{l}T:{P_2}(x) \to {P_3}(x)\\{\mathop{\rm Im}\nolimits} (T) = Sp\{ {x^3} + 1,{x^2},2{x^3} + 2{x^2} + 3\} \end{array}$$ So, $T(v)$ must be linear-dependent on those four vectors. I tried ...
2
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2answers
82 views

Equation of plane without cross product

We know that vectors $(3,3,4)$ and $(-1,-1,5)$ span a plane in $\mathbb{R}^3$. Can we somehow readily infer that the plane's equation is $x_1 - x_2 = 0$? Cross-products have not yet been introduced ...
3
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1answer
306 views

Product of vector spaces

Let $V$ be a vector space over a fixed field $k$. Under what circumstances do we have $V\times V\cong V$? I think this should be true if $\mathrm{dim} \ V=\infty$, isn't it?
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1answer
94 views

Isomorphism between 2 vector spaces

Can one vector space on $\mathbb{R}$ be isomorphic to a vector space on $\mathbb{C}$ if they have same dimension? (I was wondering about it while looking at $\mathfrak{so}(3,1)$ and ...
0
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1answer
26 views

Two points of a vector

I have a source point of a vector (x, y), the vector's size, and the angle of it. What's the formula to calculate the X and Y values of the point the vector will get to from the source point? I tried: ...
2
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1answer
126 views

Vector spaces and Scalar Multiplication

Say I have the V={set of rational numbers} as vectors and the F=field of reals as scalars. Does V form a vector space over F? I ask this because V isn't closed under scalar multiplication On ...
2
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1answer
57 views

determine basis for given vector space

let us consider following problem A subsepace $S$ of a vector space $V$ is given. Determine a basis for $S$ and extend your basis for $S$ to obtain a basis for $V$. $V=P_2$, $S$ is ...
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1answer
117 views

Why is class of one-dimensional vector spaces not axiomatizable?

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R. There are ...
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1answer
166 views

Converting from spherical coordinates to cartesian around arbitrary vector $N$

So if I'm given an arbitrary unit vector $N$ and another vector $V$ defined in spherical coordinates $\theta$ (polar angle between $N$ and $V$) and $\phi$ (azimuthal angle) and $r = 1$. How do I ...
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1answer
286 views

Prove that if transformation matrix is unitary, then the basis is orthonormal

V is a vector space with the complex field, B is an orthonormal basis of V , and C is some arbitrary basis. Prove that if the transformation matrix from basis C to B is unitary, then C is also ...
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0answers
33 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
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0answers
36 views

A question about vector function second order gradient.

I have a question about some vector function’s gradient twice, I tried for days but cant figure it please help me? Please explain it as clear as possible. Thank you very much! $\mathbf x$ is vector ...
2
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1answer
54 views

dividing a line segment in the ratio $1:2i$

The following exercise is from [Birkhoff and MacLane, A Survey of Modern Algebra]: Let $\alpha=(1,i,0), \beta=(0,1-i,2i)$. Can you ``divide the line segment $\overline{\alpha \beta}$ in the ratio ...
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1answer
78 views

Variations in math to implement three-dimensional space?

Backstory: So I was researching topics, and found that 3-D game programming often markets itself with linear algebra. As a philosopher of math I decided to dig further into this and determine if ...
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1answer
37 views

What is the dimension of this subspace?

Let $(w,x,y,z) \in \mathbb{R}^4$. Determine a basis for the subspace of $\mathbb{R}^4$ formed by the intersection of the plane $x+y-w+z=0$ with the plane $2x-y+2w-z=0$ ? What is the dimension of ...
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2answers
37 views

What is base in Linear Algebra (vector, matrix)?

For each of the following sets explain whether or not the set is/could be a basis for the space mentioned. a) $(-x,y)$ and $(x,-y)$ for $x \not = 0$ and $y \not = 0$ in $\Bbb R^2$. b) $(1,2,3)$, ...
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3answers
83 views

What’s the sum of four vectors that form a loop?

4 vectors that connectod end to end. what is the sum of all vectors ?
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1answer
53 views

Finding a vector close to vector $\vec{b}$ using $A^T$ and $A\vec{x}=\vec{b}$

I'm having a hard time understanding the rest of the steps after $A^TA\vec{x}=A^T\vec{b}$ to find $\vec{x}$ Problem: Find the vector in $W= span\ \left(\right.\ \left[ \begin{array}{c} 1 \\ 0 \\ 1 ...
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1answer
527 views

Proof: dimension of annihilator

First there is a vector space V and U is vector subspace of V. Furtermore $U^{0}$ is the annihilator of U (= {$\varphi \in V^{*} |\space\forall u \in U: \varphi(u) = 0$}). I need to show that: dim(V) ...
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2answers
64 views

Show that the functions are linearly independent

Given the following functions: $f_1(x) = e^x$, $f_2(x)=sin(x)$, $f_3(x) =cos(x)$, $f_4(x) =x$, $f_5(x) =1$ I have to show that they build a basis of a subspace of a space of all functions ...
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1answer
132 views

The sum of two forces is 18 N and resultant whose direction is at right angles to the smaller force is 12 N The magnitude of the two forces are

Problem : The sum of two forces is 18 N and resultant whose direction is at right angles to the smaller force is 12 N The magnitude of the two forces are (a) 13,5 (b) 12,5 (c) 14,4 (d) 11,7 ...
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2answers
264 views

Linear combination and Basis

Consider a set of five arbitrary 2x2 matrices. Can you always write one as a linear combination of the others? Explain. Repeat for five arbitrary 3x3 matrices. For each of the following sets explain ...
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2answers
165 views

Complements of subspaces and quotient spaces

I could use a hint on the following question Exhibit vector spaces $A$, $B$, $C$, and $D$ such that $A \oplus B = C \oplus D$, $A \cong C$, but $B \not\cong D$. I have toyed around with a few ...
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1answer
48 views

Extension of a zero linear functional

How can I show using Hahn-Banach theorem that, if $E$ is a real vector space, $F$ is a proper vector subspace of $E$, and f is the zero linear functional $f:F\to\mathbb{R}$ such that $f(x)=0$ $\forall ...
0
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1answer
137 views

Vector minus its orthogonal projection is the orthogonal projection on the complement?

Assume v is a vector of space V, U is a subspace of V .Pr is the orthogonal Projection- on the left side it is on U, and on the right side it's on the orthogonal complement of U. Is it right this ...
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1answer
23 views

How to formulate a vector of discrete data

I am trying to formulate the following english sentence: Let $v$ be a vector of dimension $9$ containing discrete data $x$ such that $x$ can only take the following values $\{-1,0,1\}$ My ...
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0answers
51 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
2
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1answer
68 views

Understanding problems of space

I've been trying to understand the concept of space for some time now, but I still can't grasp the essence of it. In high school math we've been using 2D- and 3D- Euclidean space. Now that I am ...
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0answers
22 views

Vector representation of a sum

I am trying to solve a minimization problem and I encountered the following sum: $\sum_i^n x_i^4$ I know that $\sum_i^n x_i^2 = x'x$ but I can't figure out what the vector representation is of the ...
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1answer
382 views

How to find plane equation by line and plane that perpendicular to

Find an equation for the plane that is perpendicular to the plane 2x +2y=1 and passes through the line ...
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1answer
33 views

Projection to plane definition?

What is ||Proj of U to UxV || ? no value given, so do i have to write a definition of projection? what kind of definition could be the best ? in HW Find the ...
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1answer
178 views

Is |AxB| = |A||B|sin(v) a theorem or a definition?

For cross products in $\mathbb{R}^3$, we have $ \| u \times v \| = \| u\| \| v\| \sin \alpha $. But is that a theorem or a definition? I read somewhere that it actually was a theorem which was proved ...
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2answers
133 views

About linear transformations

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
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1answer
34 views

Proof: $B$ is a basis of $V$ $\Longleftrightarrow$ $V = \bigoplus W_k$

Let $V$ be any (finite) vector space, $\oplus$ donate the (inner) direct sum, and $B_k$ is a given basis for some supspace $W_k\subset V$ and $B=\bigcup_k B_k$ then $$V = \bigoplus_k W_k ...
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1answer
116 views

Dimension of the rationals over the integers

What is the dimension of the $Q$, when they are seen as a vector space over the integers $Z$ (with the usual definitions of addition and multiplication)? Initially I thought that the dimension ought ...
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2answers
156 views

Existence of a certain subspace in a vector space

Let $V$ be a vector space, and $W_1, W_2, \ldots, W_m \subset V$ its $k$-dimensional subspaces. Every their pairwise intersection is $k-1$-dimensional (for all $i \neq j$, $\dim (W_i \cap W_j) = k - ...
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1answer
71 views

Generator of End(V)

If V is a finite-dimensional vector space of dimension n and G⊂End(V) such that G generates End(V) meaning that any element of End(V) is expressible as a linear combination of products of a number of ...