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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2answers
736 views

Formula to best fit a rectangle inside another by scaling

I am very week in Math. I am a web programmer, and usually my work does not involve too much math - its more of putting records into database, pulling out reports, making those fancy web pages etc ...
1
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1answer
48 views

$L_1 \cap L_2$ is dense in $L_2$?

We were talking about Fourier series the other day and my professor said that the requirement that a function be in $L_1 \cap L_2$ wasn't a huge obstacle, because this is dense in $L_2$. Why is this ...
2
votes
1answer
70 views

If divergence is zero, is it necessarily a curl?

The divergence of the curl of a vector is zero. But, Any vector whose divergence is zero can be the curl of a vector field ?
0
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1answer
70 views

Finding minor of the matrix

Given the following matrix: $$\begin{pmatrix} 3 & 2 & 1 & 5 \\ 8 & 5 & 8 & t \\ 2 & 1 & 6 & 6 \end{pmatrix} $$ I am looking for the matrix rank depending on ...
0
votes
0answers
50 views

On the weak* compactness of subdifferentials

Let $X$ be a normed vector space over $\mathbb R$ and $X'$ its dual space (the set of norm-continuous linear functionals on $X$). Let $f:X\to\mathbb R$ be a convex function. Consider the ...
1
vote
1answer
40 views

Implication when the product of scalar and a vector is the zero vector?

For any scalar $s$ and vector $v$ ,it holds that $sv=0$ if and only if $s=0 \ or \ v=\vec0$. A Textbook of linear Algebra gives the following proof (the reverse direction is easy to prove): Suppose ...
0
votes
3answers
52 views

Linear transformation formula

How to find formula for linear transformation $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}^4$ when the following is given: $$\varphi ((5,1))=(2,5,1,1)$$ $$\varphi((1,0))=(3,4,2,2)$$ What is the ...
0
votes
0answers
181 views

dimensions of two subspaces of a vector space not equal

I have a problem to find a relationship between two subspaces of a vector space. The two subspaces are $W_1$ which is the span of {$v_1,v_2,...,v_{n-1}$} and $W_2$ which is the span of ...
1
vote
2answers
54 views

For normed vectorspaces $V$, $A,B \subset V$ if $A$ is compact and $B$ is closed then $A+B$ is closed

I am looking for a 'direct' way to show the following statement: Problem: Let $V$ be a normed vectorspace, show that if $A$ is compact and $B$ is closed then $A+B:= \lbrace a+b \mid a \in A, b \in ...
0
votes
0answers
26 views

Projection of Vectors v on w

Given $v = [3, -6, 2]$ and $w = [-1, 6, 5]$, find; $v \downarrow w$ $w \downarrow v$ What does the magnitude of $w \downarrow v$ depend on? What does the direction of $w \downarrow v$ depend on? ...
1
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2answers
72 views

Theorem 3.6 in Sec. 3.9 in Apostol's Calculus vol. 2, 2nd edition: (How) does such a matrix exist?

Let $v_1, \ldots, v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$, the real Euclidean $n$-space, let $e_1, \ldots, e_n$ be the unit coordinate vectors in $\mathbb{R}^n$; that is, let $$e_1 ...
-3
votes
2answers
35 views

Linear combinations in a subspace of $\mathbb{R}^n$

This question is related to Prove that a subspace of dimension $n$ of a vector space of dimension $n$ is the whole space. Let $S \subset \mathbb{R}^n$ and $\{v_i\}\;( i \in \{1,2,\cdots,n\}\; v_i \in ...
2
votes
2answers
288 views

Prove that a subspace of dimension $n$ of a vector space of dimension $n$ is the whole space.

Maybe this is a stupid question. I was brought to this from the observation that an infinite dimensional vector space can have proper subspace that have the same dimension of the whole space. But, ...
5
votes
3answers
372 views

A vectorspace over an infinite field is not a finite union of proper subspaces?

Show that if V is a vector space over an infinite field F, then V cannot be written as set-theoretic union of a finite number of proper subspaces.
9
votes
1answer
104 views

$V^* \not\cong V$ if $V$ is infinite-dimensional, problem showing that $\text{Card}(\Lambda \times F) < \text{Card}\left(F^\Lambda\right)$.

Let $V$ be a vector space over $F$. Consider the dual vector space $V^* = \{f: V \to F\text{ }|\text{ }f\text{ is linear}\}$. Show that if $V$ is infinite-dimensional, then $V^*$ is not isomorphic ...
1
vote
1answer
34 views

For finite dimensional $F$-Vectorspaces $V$ it is true that $\forall U \subset V: U = \bigcap_{\lambda \in U^0}\ker ( \lambda)$

In E. Oeljeklaus & ‎R. Remmert Linear Algebra they proof this little lemma: Lemma: Let $V$ be a finite dimensional $F$-Vectorspace over a field $F$ and $U \subset V$, then $$U = ...
1
vote
1answer
16 views

Given linearly independent vectors $w_1,w_2\in\mathbb{R}^2$, $\exists\, k > 0$ such that $|m w_1 + n w_2| \geq k(|m|+|n|)\, \forall$ integers $m,n$

I am unable to see the correctness of this statement. It seems the author has considered this statement trivial and hence has not given any proof of this statement. But I am unable to prove it.
1
vote
2answers
32 views

Checking if something is a vector space

Let $C^2[0,1]$ be the set of all fucntions f such that $f'$ and $f''$ are continuous on $[0,1]$ Now we have to determine if $w = {\{f \in C^2[0,1] : f'' + 4f = 0\}}$ is a vector space with the ...
1
vote
0answers
37 views

Finding the column space of Khatri-Rao product

Let us denote the Khatri-Rao product by $\odot$. I want to find the column space of the matrix $H \odot H$, which I know has full column rank. The information is have is the column space of $H$, $c ...
2
votes
0answers
16 views

$P,Q,R$ be subspaces of a vector space $V$ such that $V=P \cup Q \cup R$ , then must one of $P,Q,R$ be equal to $V$? [duplicate]

Let $P,Q,R$ be subspaces of a vector space $V$ such that $V=P \cup Q \cup R$ , then is it true that one of $P,Q,R$ must be equal to $V$ ? I know the result about subspaces that tells that if for ...
0
votes
2answers
71 views

Proving that the range of a linear operator $A: V \to W$ is the span of the image of a basis of $V$

Let $A: V \rightarrow W $ be a linear operator and {${v_{1}, v_{2},...,v_{n}}$} be a basis of the vector space $V$. Prove that $$Range(A) = span(Av_{1},Av_{2},...,Av_{n}).$$ Let $x_{1},x_{2} \in V$ ...
1
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1answer
34 views

Linear independence in a vector space question

I'm working out the following question: If $\{u_1, u_2, u_3\}$ is a linearly independent set in some vector space. Explain why if $a_1u_1 + a_2u_2 + a_3u_3 = b_1u_1 + b_2u_2 + b_3u_3$, where ...
0
votes
1answer
55 views

How to know if a point in a circle has crossed a plane passing through the center point?

I am creating a control in .NET which computes polar coordinates based on $(x,y)$- coordinates within a panel control. Here is an image to use as a reference: When the mouse moves over the circle, ...
0
votes
1answer
31 views

Equivalence of norms in finite dimension over complete fields is true, but false for finite rank modules over complete rings

We know that if $k$ is complete valued field and $V$ a finite dimensional vector space then all norms on $V$ are equivalent. (The field is not necessarily of characteristic $0$ and its absolute value ...
0
votes
1answer
630 views

Finding the set of all points equidistant between two planes

I'm trying to study for an upcoming exam in my math class and I came across an interesting question that I'm not entirely sure about. "Let $H_1$ be the plane $x + 2y − 2z = 1$ and $H_2$ the plane $y ...
5
votes
2answers
2k views

Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vectors? My textbook is confusing about it. Any help would be appreciated. Thanks in advance!
2
votes
2answers
70 views

Complement of subspaces

Let $V$ be a finite dimensional vector space and $X$ and $Y$ be subspaces of $V$. It is obvious that if $X\subseteq Y$, then for every subspace $W$ of $V$ such that $X+W=V$, we should have $Y+W=V$. Is ...
0
votes
1answer
29 views

Span and Linearly independence of a set

Suppose that $(V, +, \cdot)$ is a vector space over a field $F$ and $S = \{v_1, v_2, \ldots, v_k\}$ is a subset of $V$. Describe the span of $S$. Explain how to determine whether $S$ is linearly ...
0
votes
2answers
23 views

If $\varphi: V \to W$ is a $F$-linear mapping, then for every $U \subset V$ it is true that $\dim_F(\varphi(U)) \leq \dim_F(U)$

Problem: Let $V,W$ be finite dimensional $F$-Vectorspaces where $F$ denotes a Field. Let $\varphi: V \to W$ be a $F$-linear mapping. Show that for every $U \subset V$ $$\dim_F(\varphi(U)) \leq ...
1
vote
1answer
95 views

Let W be an infinite dimensional vector space.Under what conditions are there only a finite number of distinct subsets S of W such that S generates W?

let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S generates W? If W is finite then obviously there only a finite ...
3
votes
1answer
23 views

Does $ f\in Sp\{f_{n}:n\in \mathbb{N}\} $?

Define $ f_{0}(x)=1,x\in [0,1) $ and $ f_{n}(x)=x^{n},x\in [0,1) $ for each $ n\in \mathbb{N} $. Also define $ f(x)=\sum\limits_{n = 0}^\infty f_{n}(x) $ for all $ x\in [0,1) $. My question : Dose $ ...
2
votes
1answer
25 views

Terminology - Union of kernels of iterated linear functions

Quick question: if $V$ is a $K$-vector space, $f : V \to V$ a linear function and $f^k = f \circ \ldots \circ f$ ($k$ times), does $\mathscr U = \displaystyle \bigcup_{k\ge 1} \ker f^k$ have ...
0
votes
0answers
52 views

Natural Transformaton $\text{Hom}(V,W)$ and $W\otimes V$

Something of this form has already been answered here: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? I'm starting introductory category theory stuff, and I'm looking for some help. I ...
0
votes
1answer
45 views

Mathematical expression for all combination of a 0-1 vector

I have a $n\times 1$ vector, $X=[x_1 ,x_2 ,x_3 ,..., x_n]$, whose elements are boolean, i.e., 0 or 1. Is there a concise mathematical expression for $\{\text{all possible outcomes of }\;X\}$? Can it ...
1
vote
1answer
36 views

Verify vector subspace - closure of addition and scalar product.

I hope this isn't a duplicate of another question but I've been trying to find something to help me and nothing has really done the job. I'm trying to verify $W$ is a vector subspace of $V$ by ...
0
votes
1answer
2k views

Given a line and a plane determine whether they are parallel, perpendicular or neither

The line $L$ passes through the point $p = (1,-1,1)$ and has direction vector $d = [ 2,3, -1]$. Determine for the plane $P$, with equation $2x+3y-z = 1$ whether $L$ is parallel, perpendicular or ...
1
vote
2answers
312 views

Find a vector that spans the given set

Question in book: Let H be the set of all vectors of the form [-2t, 5t, 3t]. Find a vector v in R3 such that H=Span{v}. Why does this show that H is a subspace of R3? Answer from solution ...
0
votes
1answer
46 views

this is not a vector space,is it?

One of my tutorial question is this $V = [0,\infty)$. For $x,y ∈ V, α ∈R,$ define $x + y = xy$, $αx = |α|x$ V is vector space or not ? Zero vector of this becomes '$1$'.and addittive inverse of ...
0
votes
1answer
42 views

Given any vector, how do you know which function space it belongs to?

One thing I cannot wrap my head around is that there are so many many many conditions for different function spaces, how can you quickly determine which function space a vector/function belongs to? I ...
1
vote
1answer
121 views

Solving a transformation equation involving vectors and quaternions

I'd like to solve the following equation for $c$, where $a$, $c$, and $d$ are position vectors represented by quaternions with $w$ (the real component) set to $0$ and $b$ is a unit quaternion: ...
0
votes
1answer
345 views

Proving subspace conditions from subsets of vector spaces

Let n>=2. Which of the conditions defining a subspace are satisfied for the following subsets of the vector space Mnxn(R) of real (nxn)-matrices? (Proofs of counterexamples needed). U={A is an ...
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0answers
60 views

To show that orthogonal complement of a set A is closed.

To show that orthogonal complement of a set A is closed. My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by ...
2
votes
3answers
200 views

Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
2
votes
2answers
145 views

$Rank(A)=$number of non-zero eigenvalues then is $Rank(A)=Rank(A^2)$?

Let $A$ be an $n$ by $n$ matrix on some field. If $Rank(A)=$number of non-zero eigenvalues of $A$ then can we say that $Rank(A^2)=Rank(A)$? I believe we can say this (thinking about idempotent ...
0
votes
1answer
138 views

Get 4 points lying on the plane by given normal

I would like to create plane using 4 points (which I need to find out), when I know the intersection point of the 2 diagonals in the plane. Next thing I know, that the Y coord of 2 bottom points will ...
0
votes
1answer
34 views

Determine the values of $k$ for which the given line and the plane are parallel

Determine the values of $k$ for which the line $\frac{x}{2}=ky=k-z$ and the plane $(2k-1)x-ky+z=5+k$ are parallel. I got the answer $k=1$ by equating the dot product of the normal to the plane and ...
-1
votes
1answer
30 views

Dependent and independent vectors.

The indexed family $u_{1},u_{2}$ where $u_{1}=u_{2} \neq \vec{0}$ are linearly dependent ( because $u_{1}$ and $u_{2}$ are collinear) and linearly independent at the same time ! we have $\alpha ...
2
votes
0answers
36 views

Question on dual spaces of vector spaces

Let $k$ be a field. Also let $E$ and $F$ be finite dimensional $k$ vector spaces. What are the most general conditions for $k$, $E$ and $F$ under which a $k$-bilinear form $\langle\;,\;\rangle ...
7
votes
1answer
134 views

Prove or disprove - Newton's method convergence in higher dimensions

It's not an exercise for uni or anything like that, just something that's been bothering me a bit and I can't seem to find useful information on the web on the matter. When talking about real valued ...
0
votes
1answer
64 views

Does $B = \{x-2, x(x-2), x^2(x-2)\}$ span $\{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$?

Let $P_3(\mathbb{R}) = \operatorname{Span} \{1, x, x^2, x^3\}$. $W$ is a subspace of $P_3(\mathbb{R})$, $W = \{p(x)\in P_3(\mathbb{R})|p(2) = 0\}$. Find a basis and the dimension of $W$. I chose ...