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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3
votes
2answers
39 views

What is the geometrical meaning of the integral of a vector valued function?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is an integrable function. then $\int_a^b f(x)dx$ can be considered as the area between the graph and the x-axis. But what if $f:\mathbb{R}^n\rightarrow \...
-5
votes
3answers
23 views

What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular? [on hold]

What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular?
2
votes
3answers
43 views

A set of linear algebra questions?

Could you help me with these questions, I figured most of them out on my own, but I'm not completely sure if I'm correct. a) $A=\begin{bmatrix}a^2&ab&ac\\ ab&b^2&bc\\ ac&bc&c^...
0
votes
0answers
23 views

Are there any differences between the mathematical definition of vectors and scalars and how they are defined in physics?

From a purely mathematical perspective, the notion of scalars and vectors and their different roles makes sense to me. Vectors are elements of a given vector space $V$, and scalars are elements of the ...
0
votes
0answers
10 views

Getting starting/endings points Related to Displacement Vector

I am using this resource to calculate the distance between two 3d line segments. At the end it provides the 3D Vector dP. The length of this vector provides the correct distance between the two lines ...
0
votes
0answers
10 views

Locally convex vector space and balanaced sets

Let $f : \mathbb{R}^{n \times d} \times \mathbb{R}^d \longrightarrow \mathbb{R}$ be the following function: $$f(P,\theta) = \left| \left| \theta \right| \right|_2 + \ln{(1+e^{P_i\theta})} ,$$ where $...
0
votes
0answers
14 views

Getting points Related to Displacement Vector

I am using this resource to calculate the distance between two 3d line segments. At the end it provides the 3D Vector dP. The length of this vector provides the correct distance between the two ...
0
votes
0answers
11 views

Characterizing spaces in which Cauchy-Schwarz holds

Let $V$ be a vector space over $\mathbb R$ and $g:V\times V\rightarrow \mathbb R$ be a bilinear form on $V$. The usual statement for the generalized Cauchy inequality usually goes like this: Let $...
2
votes
0answers
35 views

Why are vectors portrayed as a sub space of $R^n$

Typically I have seen all vector notation showing it as a sub space of $R^n$. Why not $Q$ or $Z$?
0
votes
1answer
39 views

Let V subspace of W, with dimV=dimW. Why should they be equal?

The fact that $V$ is a subspace of $W$ , means $\dim V \leq \dim W$ . We are told $\dim V = \dim W.$ So if $B_1$ is the basis of $V$ and $B_2$ is the basis of $W,$ it is $ |B_1|= |B_2| .$ We know that ...
1
vote
1answer
39 views

Showing that a function in a vector space is linear

Let $X$ be a vector space and consider a function $f : X \rightarrow \mathbb{R}$ defined for some $a \in X$ defined as $f_a (x) = a \cdot x$. (i) Prove that $f_a (x) = a \cdot x$ is a linear function....
2
votes
2answers
64 views

Two vector spaces with same dimension and same basis, are identical?

Let $V$ subspace of $W$ and both have same dimension and same basis. Then can we safely say that $V= W$ ? I believe yes. For example there may be an element $x \in V$ written as a linear combination ...
3
votes
2answers
44 views

Find m so that $(m+1,1,1)$ , $(1,-m,-1)$ , $(m,1-m,2)$ are linearly dependent

I formed an augmented matrix $$\left(\begin{array}{ccc|c}m+1&1&m&0\\1&-m&1-m&0\\1&-1&2&0\end{array}\right)$$ I now that we do reduced row echelon form for the ...
0
votes
1answer
37 views

Normalised Basis for vector space V.

Let $$ V= \{ (x_{1}, x_{2}, x_{3})' \in \mathbb{R}^{3} | \, 3 x_{1} + x_{2} = 0 \text{ and } 2 x_{1} - x_{3} = 0\} $$ What is the normalised basis for V? I tried it two different ways: $x_{2} = -3 ...
0
votes
0answers
47 views

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

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 calculating ...
0
votes
2answers
68 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
votes
2answers
18 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
37 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 ...
1
vote
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
votes
0answers
9 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
vote
0answers
37 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 ...
3
votes
1answer
37 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 ...
2
votes
1answer
30 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
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\...
0
votes
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} \...
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
votes
1answer
41 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 ...
0
votes
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^...
1
vote
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{...
-1
votes
0answers
29 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 \...
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
0answers
23 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
vote
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
votes
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
votes
1answer
25 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
votes
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
88 views

'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 ...
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,(\...
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 ...
4
votes
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
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 =...
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=\...
1
vote
0answers
17 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
32 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-...
-3
votes
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
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
0
votes
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 ...