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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2
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1answer
303 views

Find equation of line such that area formed by line & positive coordinate axis is minimal

Find equation of line passing through $(20,12)$ such that the area of the triangle formed by the line and the positive axis is smallest possible. Also: $\frac{x}{a}+\frac{x}{b}=1$ where $a, b$ are ...
0
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2answers
217 views

What is the use of a transpose of a matrix in an equation and how to solve one?

I have the following equation to solve, $$g(x) = x^t W_i x + {W_i}^t x + v_{i_0}$$ In this equation why the need to use a $x^t$ and $x$? I feel $x$ and transpose of it both are the same ($x$ is a row ...
0
votes
2answers
79 views

Proving $(f^{-1}(U))^0 = f^*(U^0)$

Let $V, W$ be two finite-dimensional vector spaces, $f: V\rightarrow W$ a linear map, and $U \subseteq W$ a vector subspace. I'm trying to show that $(f^{-1}(U))^0 = f^*(U^0)$, i.e. that the ...
0
votes
1answer
239 views

Eigenvector corresponding to zero eigenvalue / identical eigenvalues, not-identical eigenvectors

Assume symmetricmatrix $B\in\mathbb{R}^{n\times n}$ is given, and a transformation $$A=JBJ,$$ where $J=I - \frac{1}{n}1_n1_n^T$ and $I$ denoting the identity matrix, hence centering its rows and ...
0
votes
1answer
109 views

Nullspace of combination of two basis vectors

I have two 6x1 element basis vectors $s_1$ and $s_2$ defined in local coordinates which can be combined into a single 6x2 subspace of $$ s = \begin{vmatrix} X_{2}s_{1} & s_{2}\end{vmatrix}$$ ...
0
votes
0answers
118 views

Linearity property of Mathematical objects

I can't help notice that linearity property is rather common among various mathematical objects like finite dimensional vector spaces and functionals, Expected Value of random variables. I am sure ...
0
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3answers
111 views

System of 4 equations (3 of them linear) - What can be said about the solution set

I have a homework at linear algebra and we have this system of linear equations: $ x+y+z+w=0 $ $ x+2*y+9*z+13*w=0 $ $4*x+41*y+6*z+656*w = 0 $ And we add this equation: $ x^3 + y^4 + 8* z^5 + 8* ...
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0answers
135 views

Magnetic vector potential

I have a magnetic field component of the form $B_{\phi} = z \, \sin ^4 (r/a) \, r/a$, and $B_z = b$, I need to evaluate magnetic vector potentials. As we know, $B = \nabla \times A$. How to solve ...
6
votes
1answer
199 views

When does there exist an isometry that switches two subspaces?

Let $V$ be a real vector space of finite dimension and let $\langle \cdot, \cdot \rangle$ be a non-degenerate symmetric bilinear form on $V$. Let $U, W \subseteq V$ be linear subspaces such that ...
0
votes
1answer
114 views

Show that (vector) subspaces of $\mathbb{A}^n$ are algebraic sets

i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of ...
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2answers
181 views

An example of an incomplete vector space [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? I am looking for an example of a set and operations on this set that isn't quite a ...
0
votes
1answer
52 views

Orthogonal vector with fixed lenght

Given a long vector called A and a direction vector called B, how can one retrieve a position vector OC where a orthogonal line casted down to A has a fixed lenght? The image above shows a 2d ...
5
votes
6answers
449 views

Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis?

Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis? I solved a similar problem in $\mathbb{R}^2$ (or at least think I did :p). ...
7
votes
2answers
869 views

Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
0
votes
1answer
137 views

Solving Vector Problem given Point & Line

I need to find the equation of the plane that contains point $(1,0,0)$ and line $r=(1+\lambda)i + 3j + 2\lambda k$ Correct Answer Plane parallel to $(1,3,0)-(1,0,0)=(0,3,0)$ Plane perpendicular to ...
11
votes
4answers
3k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
2
votes
1answer
722 views

Finding the angle between u and v.

Struggling with the following Suppose that $u$ and $v$ are non-parallel unit vectors, $a=u+sv$ and $b=u-sv$, where $s$ is a real number. If the angle $\theta$ between $u$ and $v$ is the same as the ...
1
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2answers
2k views

Vector Projection XY plane

How do I find orthogonal projection of a vector $\vec V_1=(2,3,4)^T$ formed with the points $A(0,0,5)$ and $B(2,3,9)$ on $xy$ plane?
3
votes
1answer
111 views

A vector calculus question

I realize that this sounds like a physics question, but what I am stuck on is a mathematical issue, so I hope you won't mind me posting this question here. I have a cylinder given by the equation ...
4
votes
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
164 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
702 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
vote
1answer
872 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, ...
1
vote
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
420 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
977 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
563 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
333 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
733 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
179 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 $I$ ...
1
vote
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
686 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 ?
1
vote
1answer
526 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
970 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
185 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
244 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
648 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) ...