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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5answers
2k views

What is the difference between metric spaces and vector spaces?

Does a metric space have an origin? That is, does it have $(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this ...
1
vote
1answer
163 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
2
votes
3answers
146 views

Explanation to the details of the proof that $F[x]$ is not finite-dimensional.

I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor. The space of polynomials $F[x]$ is ...
3
votes
2answers
699 views

Extend angle between two 3D vectors to x-y plane.

I would like to know how I can extend the angle between two vectors in 3D space to the x-y plane. So, there are two vectors in 3D space, and the angle between them is found using the definition of ...
8
votes
2answers
6k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\textbf{x}[m] = ...
2
votes
0answers
2k views

How can I find two vectors in a given span {u, v} that are not multiples of u or v?

But do appear to be linear combinations? $u$ and $v$ are 3-component vectors. The question posed is: Find two vectors in span{u,v} that are not multiples of u or v and show the weights on u and v ...
1
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1answer
867 views

What is the relationship between spans that contain some of the same vectors?

I have been given the following problem: Let x, y, z be non-zero vectors and suppose w = 4x + y -3z. a) If z = 4x + y, then w = _x + _y. b) Using the calculation in (a), mark the ...
0
votes
0answers
39 views

A subset that can be scaled to be the whole space, or that can contain a scaled version of the whole space

The following is from Mariano's comments on my earlier question In a topological vector space, why is the following true: if a neighborhood U of zero contains a scaled copy of the whole space, ...
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1answer
580 views

Sketching a line segment from a vector equation

Sketch the line segment represented by each vector equation: $$\begin{align} r &= (1-t)(i+j) + tk \;& 0 \le t \le 1 \\ \\ r &= (1-t)(i+j+k) + t(i+j) \;& 0 \le t \le 1 ...
1
vote
1answer
418 views

Angle between functions

I have a rather simple question but googling it did not bring a satisfactory result: Assume you have given two function $f$ and $g$ on some space $\mathcal{L}^2(\Omega)$ where $\Omega \subset ...
3
votes
1answer
976 views

dual space is a vector space

I was wondering if some one could please shed some light on why or how a dual space itself becomes a vector space over the Field. The "Finite Dimensional Vector spaces" book by Paul Halmos states."To ...
0
votes
1answer
561 views

line projection on top of a plane

If I have a horizontal line (a 3d point and 3d vector with zero z component) and another plane (could be an oblique or a horizontal; i have normal vector of the plane); then how do we get the ...
0
votes
1answer
149 views

How to find the trace of differentiation operator on a vector?

I need to find the trace of differentiation operator $D$ on a polynomial vector space $P$ with degree $n$. $Dp(x)=p'(x)$. According to wikipedia trace can be found by representing the basis in ...
3
votes
3answers
190 views

Possible proof for the relation involving matrix trace

Suppose a diagonal matrix $D\in\mathbb{R}^{n\times n}$ is given, with all its entries $d_{ii}\geq0$, for all $i$. Is it possible to prove ...
0
votes
1answer
330 views

What is a general scalar and what a (complex conjugate)

I've been reading something about Quantum Mechanics where they introduce the maths slightly more rigorously. They talk about vector spaces and an inner product which yields a scalar. Moreover complex ...
2
votes
2answers
201 views

2 linear functionals on a vector space so one can be represented as a multiple of the other.

Prove that if $y$ and $z$ are linear functionals (on the same vector space) such that $\left[ x,y\right] = 0$ whenever $\left[ x,z\right] = 0$, then there exists a scalar $\alpha $ such that $y=\alpha ...
2
votes
2answers
731 views

Positive semi-definite matrix

Suppose a square symmetric matrix $V$ is given $V=\left(\begin{array}{ccccc} \sum w_{1s} & & & & \\ & \ddots & & -w_{ij} \\ & & \ddots & ...
0
votes
1answer
47 views

Is $y\left( x\right) =\dfrac {d^{2}x} {dt^2}|_{t=1}$ a linear functional for vector space of polynomials?

Let $P$ be the set of all polynomials, with complex coefficients, in a variable $t$. For $x$ in $P$ the function $y$ is defined by $y\left( x\right) =\dfrac {d^{2}x} {dt^2}|_{t=1}$ Is $y$ a linear ...
2
votes
1answer
178 views

Elementary vector calculus: Divergence of a field

How do you find the divergence of a field $f(\vec{r})=\vec{r}\exp(r^2)$ where $\vec{r}$ is the position vector and $r$ is its magnitude? In other words, how does one evaluate ...
1
vote
2answers
97 views

Understanding the structure of a finite dimensional vector space based on the properties of linear maps to itself

Let $V$ be a finite dimensional vector space over $\mathbb{R}$. What can we say about the dimension of $V$ if we know that there exists some linear map $\phi: V\to V$ such that $\phi^n=-I$, where $I$ ...
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1answer
176 views

Finite dimensional vector space with subspaces [duplicate]

Possible Duplicate: Could intersection of a subspace with its complement be non empty. Is it possible for a finite dimensional vector space to have 2 disjoint subspaces of the same ...
0
votes
2answers
684 views

Can a Vector space have subspaces of same dimension over different fields?

Just wondering if a finite dimensional vector space could have two subspaces such that each of these subspaces has the same dimension but form vector spaces over different fields ?
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1answer
523 views

Could intersection of a subspace with its complement be non empty.

If that is possible could you please correct my understanding about complement of a subspace. From what i recall from set theory. A complement of a set B is the set U - B where U is the universal ...
1
vote
1answer
191 views

Counter example for a result of intersection of subspaces

I am struggling with this question from Halmos's text, please ignore the imperative language. "Suppose that $L, M$ and $N$ are subspaces of a vector space. Show that the equation $$L \cap (M + N) ...
4
votes
1answer
155 views

What does a subspace spanned by another subspace and a vector mean?

What does a subspace say A spanned by another subspace B and a vector x mean ? Does that imply anything about a basis or does it just mean that every vector in subspace A is either present in ...
0
votes
1answer
967 views

Finding a result vector from 2 vectors without cross product

If I have 2 lines with its symmetric equations I can get the vectors U and V of each line, and with a cross product I can get the vector R; but how can I get the vector R without a cross product?
0
votes
1answer
139 views

Vector subspaces are the same as the parent vector space.

I came across this question in Halmos's book which i was not sure how to answer "If M and N are subspaces of a vector space V, and if every vector V belongs either to M or to N(or both), then either ...
4
votes
1answer
199 views

How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help ...
0
votes
2answers
184 views

Are two rational vector spaces having the same cardinal number are isomorphic?

Discuss the following assertion:if two rational vector spaces have the same cardinal number(i.e., if there is some one-to-one correspondence between them), then they are isomorphic(i.e., there is ...
1
vote
1answer
243 views

Number of vectors in an n-dimensional vector space.

How many vectors are there in an $n$-dimensional vector space over the field $\mathbb{Z}/(p)$ (where $p$ is prime)? Would the answer be $p^n$?
3
votes
0answers
646 views

Three-dimensional vectors and force systems

Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below As shown, a system of cables suspends a crate weighing W = 350 . (Part C 1 figure) ...
1
vote
2answers
209 views

Can you combine axioms for a vector space?

This is what I wrote...initially I wanted to write that it is false because an axiom is a basic property and wouldn't be so basic if you start combining them. In the axioms for a vector space, can ...
3
votes
1answer
215 views

Orthogonal matrix, translations, and fixed points

Is it true that $Ax+b$ where $A^\dagger A=I=AA^\dagger$ and $A$ is an $n\times n$ real matrix, $x,b\in \mathbb R^n$ must have either a fixed point or a fixed $n-1$ hyperplane? If not, is it true for a ...
1
vote
1answer
251 views

How to get a projected 3d line segment, lie on another 3d line parallel to that line segment.

I have a 3D line segment and another 3D position which locate slightly away from the line segment. I want to get the projected line segment (3D) which lies on imaginary 3D line which passes through ...
1
vote
1answer
290 views

Doubt with parametric and symmetric equations

In the line through $P(0, 0, 0)$ and is perpendicular to $x=y-5$, $z=2y-3$, when we solve the equations and get the symmetric equations in order to find the vectors $V_1$ and $V_2$, why the normal ...
1
vote
1answer
50 views

Rescale second vector by normalization transformation of first

First post, let's hope you don't laugh. I have a program that does 1,000 trials to calculate two values, a score and the std deviation of that score. So now I have two columns of values The first ...
0
votes
1answer
56 views

what is the column space for 2-d vectors

If i have two vectoes [1,2] and [3,7] what will be the column space ? Will it be a plane or a line ? In the case of [1,2] and [3,6] it is the line y=2x . Can the column space for 2-d vectors be a ...
0
votes
1answer
142 views

Proof of conjecture which yields bases of vector Space $\mathbb{C}$ over Field $\mathbb{Q}$

I am trying to determine whether the following does or does not create bases of vector Space $\mathbb{C}$ over Field $\mathbb{Q}$. define an equivalence on $\mathbb{C}$ by $x\sim y$ iff there exists ...
1
vote
0answers
63 views

Is the associativity of vector addition a consequence of the others conditions in the definition of a vector space? [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? I know that the commutativity of vector addition is a consequence of the others ...
5
votes
1answer
76 views

Extensions of finite-rank operators

Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the ...
3
votes
1answer
292 views

Maximum cosine for angle between 2 vectors when 1 vector is partially unknown

assuming I have two vectors $A$ and $B$, where $A$ is completely known and from $B$ I know only that the first k components are 0. What is the maximum possible cosine value for the angle between the ...
1
vote
2answers
541 views

Proving that if $a\mathbf{x}=\mathbf{0}$ then $a=0$ or $\mathbf{x}=\mathbf{0}$

Prove that if x is a vector and a is a scalar, then the following relation holds ? 1) if ax = 0, then either a = 0 or x = 0 ( or both). This is trivial although i am unsure if my steps are correct. ...
3
votes
3answers
3k views

Complex vector spaces

I was wondering if someone could please shed some light on how the following two vector spaces are different. The examples are from Paul Halmos's "Finite Dimensional vector spaces" book and the author ...
2
votes
4answers
2k views

How to test any 2 line segments (3D) are collinear or not?

if we have two line segments in 3D, what would be the way to test whether these two lines are collinear or not? (I fogot to mentioned that my line segments are 3D. So, I edited the original post. ...
7
votes
2answers
2k views

Cosine similarity / distance and triangle equation

There is a similarity function particular popular for processing sparse vectors such as textual data (word frequency counts etc.) commonly referred to as cosine similarity. There are two variants to ...
4
votes
1answer
175 views

Vector subspace decomposition problem (Linear algebra)

I'm stuck with the following problem and I've tried approaching it by extending the initial base of $W$ without luck. Any hints?? Consider two subspaces $W_1$ and $W_2$ of the vector space $\mathbb ...
2
votes
1answer
157 views

Define two differents vector space structures over a field on an abelian group

Exercise 3 from Roman's book "Advanced Linear Algebra". The author asks us to "find an abelian group $V$ and a field $\mathbb{F}$ for which $V$ is a vector space over $\mathbb{F}$ in at least two ...
0
votes
1answer
356 views

Computing the average coordinate for more than 2 points on a 3d line segment

Suppose, I have many 3d line segments which suppose to be intersected with another given line segment. So, I wish to take a line segment and the given line to get the intersection point. Again, I wish ...
6
votes
3answers
263 views

Is $\mathbb{Z}(p^{\infty})$ a vector space over some field $\mathbb{F}$?

I don't know how to write in good English, so I will follow Hungerford's word from his book Algebra. The following relation on the additive group $\mathbb{Q}$ of rational numbers is a congruence ...
3
votes
1answer
312 views

clarification on the definition of direct product of vector spaces

In the Roman's book (Advanced Linear Algebra) he defines the direct product of a family of vector spaves over $\mathbb{F}$ as follows: Definition: Let $\mathcal{F}=\{V_{i}| i\in K\}$ be any family ...