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 ...

learn more… | top users | synonyms

1
vote
1answer
7 views

Finding The Shortest Distance Between Two 3D Line Segments

I have two Line Segments, represented by a 3D point at their beginning/end points. Line: ...
0
votes
1answer
18 views

Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
0
votes
0answers
8 views

Proving set of upper triangular matrices form subspace of $M_{m \times n}$ (F)

Proving set of upper triangular matrices form subspace of $M_{m \times n}$ (F) Now clearly if i set all elements including diagonal =0 then we can say that o vector is there. Also it satisfies ...
1
vote
1answer
28 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
3
votes
2answers
53 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 \...
4
votes
0answers
612 views

Show that if $S$ is a subspace of a vector space $V$, then $\dim (S) \leq \dim (V)$

Show that if $S$ is a subspace of a vector space $V$, then $\dim(S) \leq \dim(V)$. Furthermore, if $\dim(S)= \dim(V) < \infty$ then $S=V$. Give an example to show that the finiteness is required in ...
6
votes
1answer
266 views

Subspace generated by permutations of a vector in a vector space

Let $K$ be a field. Consider the vector space $K^n$ over the field $K$. Suppose $(a_1,a_2, ... ,a_n) \in K^n$. What is the dimension of the subspace generated by all the permutations of $(a_1,a_2,...,...
-4
votes
3answers
27 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
46 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
2answers
1k views

Change of basis matrix for polynomials?

I've understood what a change of basis matrix is, and how it's structured. So a change of basis matrix from $B$ to $C$ is the matrix $M$ such that: $${\begin{bmatrix} &\\ \\ \\\end{bmatrix}}_B =...
0
votes
0answers
25 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
13 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 $...
3
votes
0answers
90 views
+500

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
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 $...
1
vote
1answer
41 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
0answers
36 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 ...
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 ...
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
45 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
38 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
1answer
389 views

Finding a basis for a particular subspace with Dot Product restrictions

Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then,...
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
38 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
votes
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
541 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
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
2answers
498 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
vote
0answers
38 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
votes
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
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\...
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 ...
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 ...
2
votes
1answer
31 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
votes
0answers
21 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
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
votes
0answers
30 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
votes
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 +...
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 ...
0
votes
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
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{...
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
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)}\...