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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2answers
50 views

Is the finite dimension of a vector space over the complex numbers half the dimension of the same vector space considered over the reals?

Consider a vector space V with basis ${b_{1},..,b_{n}}$ and complex scalars. This obviously has dimension n. Now consider a space with the same exact set of vectors of V, except with real scalars. I ...
0
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0answers
23 views

How is the problem of “comparing two vectors with the same dimensions but different number of elements” called?

I think that there is a specific name for the problem of "comparing two vectors with the same dimensions but different number of elements". Maybe also in the context of comparing sets or ...
0
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2answers
17 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
0
votes
1answer
36 views

Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
2
votes
5answers
153 views

How to prove $ A^{\perp} $ is a closed linear subspace?

Suppose $ X $ is an inner product space and $ A\subseteq X $. I need to prove that $ A^{\perp} $ is a closed linear subspace of $ X $. Can anyone give me a idea?
0
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1answer
47 views

Is there a linear transformation that sends v to (1,0)

Suppose we are given a linear vector field $v(x) = Ax$ on $\Bbb R^2$. Suppose $x_0$ is the point where this vector field does not vanish. Is it possible to find a linear transformation from $\Bbb R^2$ ...
3
votes
3answers
539 views

Space spanned by matrices

I have a set of $5 \times 5$ matrices, $M_1, M_2,\dots, M_{19}, M_{20}$. 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 ...
1
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1answer
36 views

Algebraic number spaces

While studying about Vector spaces and subspaces I came across the following question:- $Q.$ Do $algebraic$ numbers form a subspace of the vector space $\Bbb R$? According to my knowledge of $...
0
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0answers
8 views

when is a separable vector a product vector?

Consider a real tensor product space $V^{(1)}\otimes V^{(2)}$, and a set of vectors of the form $a\otimes b$. A "product vector" is defined as one that separates over the tensor product, e.g. $(a+b)\...
1
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2answers
491 views

Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
1
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0answers
34 views

$V$ be a vector space , $T:V \to V$ be a linear operator , then is $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$? [duplicate]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$ ? (note that the direct product is well-defined as both the spaces ...
0
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1answer
1k views

Find the vector equation of a line passing through point A perpendicular to line AB

'Points A and B have coordinates (4,1) and (2,-5) respectively. Find a vector equation for the line which passes through the point A (only the point A), and which is perpendicular to the line AB.' ...
-1
votes
1answer
49 views

Dimension of kernel for nilpotent transformation powers

Let $T:\Bbb R^n \to \Bbb R^n$ be a nilpotent transformation with index $n$. (i.e. $T^n=0$). Is it true that for all $n≥k≥0$, $\dim \ker T^k=k$? How can that be shown? The context is a linear algebra ...
1
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2answers
35 views

Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: \begin{equation} h = \sum_n \langle h,h_n\...
3
votes
1answer
36 views

$V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2) $?

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2) $ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement ...
0
votes
1answer
40 views

Intuitive way to understand the use of matrix inversion to find dual basis

I'm currently thinking about the following problem: Problem: Let $B = (b_1, b_2, b_3)$ a base of $\mathbb{R}^3$. Find the correlating dual basis $B^* = (b_1^*, b_2^*, b_3^*)$. $B$ is explicitly ...
2
votes
1answer
28 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
1
vote
1answer
34 views

linearly dependent family of vectors.

Can someone help me to solve this question please : Establish, by induction, that : $ \forall n \in \mathbb{N} \setminus \{ 0,1 \} \ \forall v_1 , \dots , v_n \in \mathbb{R}^n $ linearly independants ...
0
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0answers
20 views

Is there an algorithm for finding the largest possible linear subspace of a given vector space having this specific property?

Let $G_1,G_2,\dots,G_k$ be $n\times n$ real matrices, and let $\mathcal{G} = \operatorname{span}\left\{ G_k\right\}$. Let $\mathcal{V}$ be a linear subspace of $\mathcal{G}$, i.e. $\mathcal{V} \...
0
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1answer
20 views

Definitions of intrinsic core of convex set

Let $C$ be a convex subset of a vector space $V$. We consider two definitions of the intrinsic core of $C$. Definition 1. The intrinsic core of $C$ consists all points $c\in C$ such that for every $c^...
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0answers
25 views

Definition of space by convex function [closed]

It is well know that it is possible to define a space by norm, e.g. lets say that the norm we are concentrating on is L3 norm, thus $C = \{\theta \in \Re^d \mid \| \theta \|_3 \leq 1\}$ where $d \in \...
3
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1answer
5k views

How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i +...
2
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0answers
59 views
+50

'Tetrahedral' coordinates in space (generalization of hexagonal coordinates)

The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal. Imagine ...
0
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2answers
4k views

Linearly Independent set of vectors that spans the same subspace of $\mathbb{R}^3$

I'm having trouble setting this up. I have these $3$ column vectors: $\langle 1, 1, 2\rangle$ $\langle -7, -1, -8\rangle$ $\langle 3, 0, 3\rangle$ I need to find a linearly independent set of ...
1
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1answer
24 views

Why are not these two sets subspaces of $\mathbb{R}^3$?

Why are not these two sets subspaces of $\mathbb{R}^3$? $$ \begin{align} S_1&=\left\{\begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}:x_1=x_3\text{ or }x_2=-2x_3 \right\}\\ S_2&=\left\{\begin{...
2
votes
1answer
44 views

Proving/verifying dimension and basis

I'm coming from a computer science background and am currently trying to formalize my linear algebra knowledge by going through Linear Algebra Done Right. I have an intuitive grasp on most of the ...
0
votes
1answer
22 views

A basis for a tensor product space where the tensor elements are linearly dependent

Say I have a space $V^{(1)}$ with basis $\{a_i \}$ and $V^{(2)}$ (with dimensions $d_1$, $d_2$ respectively) with basis $\{b_j\}$. Clearly the vectors $\{a_i\otimes b_j\}$ are a basis for $V^{(1)}\...
0
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0answers
22 views

Names for the vector spaces $T(V)$ and $S (V)$

Are there any names for the vector spaces $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and $S(V)= \bigoplus_{n\geq 0} V^{\otimes n}/\Sigma_n$? The best thing I could come up with is "the underlying ...
1
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2answers
39 views

Is it correct this reasoning?

Let $E,F$ be reals vector space. Since (1) $\dim (E\times F)=\dim E + \dim F$ (2) $\dim\ \text{Hom}(E,F)=\dim E\cdot \dim F$ Given $r>0$ integer, is it true that: $$\text{Hom}(E\times \stackrel{(...
0
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0answers
14 views

$Hom(E\times\stackrel{(r)}{\ldots}\times E,E)$ isomorphic to $\bigotimes_r^1 E$?

Let $E$ be a $n$-dimensional $\mathbb{R}$-vector space. Prove that: $$\begin{array}{ccll} \Phi:&Hom(E\times \stackrel{(r)}{\ldots} \times E,E)&\longrightarrow&\bigotimes_r^1 E\\ &\psi &...
0
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1answer
37 views

Hermitian adjoint

I'm trying to solve this task, but I'm not sure, if my solution for a) is correct. For b), i dont find a starting point. Did someone have an idea how to solve this? Thanks in advance. Be $V$ the set ...
-4
votes
0answers
28 views

A=kB where K is a scalar [closed]

suppos that A is a square matrix. in each case below, state what you know about det(A). A. det (A^1)= 5 B. Ax= b represents a consistent system of equation. C. if A is 3x3 and det(2A)=8 d A=kB ...
1
vote
2answers
16 views

Find the parameter $\lambda$ such that the dimension of a vector subspace is equal to $2$.

Given the set of vectors $\{a,b,c\}$ in $\mathbb{R^3}$ that is linearly independent. Determine the parameter $\lambda\in\mathbb R$ such that the dimension of a subspace generated by vectors $2a-3b,(\...
4
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0answers
39 views

scalar product for $\mathbb Q$-vector space in $\mathbb C$

In the texbooks I have for linear algebra, the scalar products are only introduce for $\mathbb R$ and $\mathbb C$ vector space, that lead me to following question: $W:=span(1,\sqrt{2}) \subset \...
0
votes
0answers
25 views

Calculate the adjoint map

i'm trying to solve this task, but I don't find a starting point. Did someone have an idea how to solve this? Be V the set $\{f \in \mathbb{R}[X]| grad\,f \leq 2 \}$. This becomes to an euclidic ...
0
votes
1answer
17 views

question about basis and norm( conception and computation)

This is a multiple-choice question. I think the first choice is correct, because the x is the coefficient and $\phi_i$ is the basis. And I think the third is false because $\parallel\phi_i\parallel_2 =...
2
votes
5answers
1k views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
0
votes
0answers
14 views

Counterintuitive Property of High-D Vector Spaces

Is it possible, and if so how, can two pairs of sets of points in a vector space be constructed so that each pair has a one-to-one mapping between the sets, and where the mappings satisfy the ...
3
votes
3answers
59 views

Find vectors that span the kernel of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$

I have the following matrix: \begin{bmatrix}1&2\\3&4\end{bmatrix} and I'd like to find the vectors that span the kernel. The book I'm reading isn't helping me understand this concept at ...
0
votes
1answer
23 views

$\{x_1,x_2\}$ linearly independent. $\{x_1,x_2,u,v\}$, $\{x_1,x_2,w,z\}$ are basis => $\{u,v,w,z\}$ not a basis?

Let $V$ be a vector space of $\dim(V)=4$, and $\{x_1,x_2\}$ are linearly independent in $V$. We can complete it to a basis of $V$: $B_1=\{x_1,x_2,u,v\}$ and another one $B_1=\{x_1,x_2,w,z\}$. Is $C=\...
41
votes
14answers
76k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
1
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0answers
16 views

Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...
0
votes
1answer
10 views

Finding an equation for a plane passing through three points (Serge Lang Example Problem)

An example problem from Serge Lang's Calculus of Several Variables (pg. 30-31): Example 3. Find the equation of the plane passing through the three points $$ P_1 = (1,2, -1), P_2 = (-1, 1, 4),...
0
votes
1answer
31 views

Concerning the Gilbert Strang's book about algebra and the special solution of the nullspace.

Unfortunately I don't have yet 10 reputation, so I can't post the pic from the book, so I will paste the link. https://s32.postimg.org/g8divtz6t/Screen_Shot_2016_07_15_at_01_56_24.png My question is-...
0
votes
2answers
50 views

evaluating curl of $\vec r/r^2$

how do I calculate curl of : $\vec r/r^2$ I don't know how to solve this problem can someone help me please
-3
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1answer
62 views

help with vector calculus [closed]

the question is : how do I prove that: $\nabla^2 (r^n\vec r)=n(n+3)r^{n-2}\vec r$
0
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0answers
30 views

How to define a transported version of the simplex $X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$?

Let the simplex in $\mathbb{R}^n$ be denoted as $$X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$$ So it looks like: I want to take a point $\bar x \in X$ (red dot) And drag it to ...
0
votes
1answer
43 views

Which vectors are obtainable by my function?

Imagine a disc with $N$ radially displaceable masses $m_g$. A total imbalance with respect to the center of the disc can be calculated as follows (using the respective radiuses $r_1,...,r_N$): $$\...
1
vote
1answer
20 views

Matrices always permute a vector space basis?

Is it possible that given any matrix in $GL_n(K)$ (with $K$ a field) where the selected matrix has finite order, one can show a basis that is permutated by the transformation made by the selected ...
2
votes
1answer
37 views

Can $f(x,z) = x^Tx + \sum\limits_{i = 1}^n \dfrac{x_ix_i}{z_i}$ be written with multiple inner products at the same time?

I am running into a very interesting phenomenon that I do not quite understand (Illustration of an example of so called subset of $\mathbb{R}^n$) For example, suppose we have a subset of $X \...