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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161 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 ...
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2answers
50 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 ...
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0answers
25 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? ...
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2answers
70 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 ...
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2answers
33 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
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2answers
230 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, ...
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3answers
271 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.
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1answer
97 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 ...
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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 = ...
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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.
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2answers
31 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 ...
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0answers
29 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
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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 ...
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2answers
69 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 $x_{1},x_{2} \in V$ ...
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1answer
32 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 ...
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1answer
53 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, ...
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1answer
29 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 ...
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1answer
548 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 ...
2
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2answers
67 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 ...
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1answer
27 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 ...
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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 ...
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1answer
83 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 ...
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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
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1answer
22 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 ...
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0answers
48 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 ...
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1answer
44 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 ...
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1answer
31 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 ...
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1answer
806 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 ...
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2answers
159 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 ...
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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 ...
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1answer
41 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
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1answer
77 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: ...
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1answer
246 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
50 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 ...
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3answers
150 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
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2answers
106 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 ...
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1answer
118 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 ...
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1answer
26 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 ...
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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
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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
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1answer
121 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
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1answer
62 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 ...
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1answer
46 views

How to understand “completeness” intuitively?

In my text, it says, "if cauchy sequence in a normed vector space converge, i.e. $$\lim_{j,k \to\infty} ||u_j - u_k|| = 0$$ then the normed vector space is complete". The definition of completeness ...
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0answers
41 views

Which of the following vector spaces are isomorphic? [on my last try]

So far I have tried for the first problem: A&B, B&C, A&B&C and for the second problem B&C&D, A&B&C&D, & A However, they have turned out to be wrong. I know ...
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2answers
26 views

What is the dimension of $Graph(T)$?

Let there be $T:\mathbb{F^n}\rightarrow \mathbb{F^n}$ a linear transformation, and $Graph(T):=\{\,(v,T(v))\mid v\in \mathbb{F^n}\,\}$. What is $\dim(Graph(T))$? The answer is $n$ but if $Graph(T)$ ...
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4answers
44 views

If two invertible matrices agree on a vector, does this imply their determinant agrees as well?

As stated, if we let $A, B \in M_n(\mathbb{R})$ be invertible and there is some $v\in R^n$ such that $$Av = Bv$$ does it follow that $\det(A) = \det(B)$? Additionally, does this hold if we let $A, B ...
0
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1answer
32 views

Converting $\mathbb{C}$ linear tranformation with determinant $a+bi$ into an $\mathbb{R}$-linear transformation with determinant $a^2+b^2$.

Let $V=\mathbb{C}^2$. Let $T:V\rightarrow V$ denote a $\mathbb{C}$ linear tranformation with determinant $a+bi$, $a,b\in \mathbb{R}$. Prove that if we regard $V$ as a $4-$dimentional real vector ...
2
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1answer
36 views

Finite dimensional vector space $V$ and $\operatorname{End}_k(V)$.

This is a homework problem. I want to solve it independently as best I can, so please only give awesome hints. Let $k$ be a field. Let $V$ be a vector space over $k$. I want to prove that $V$ is ...
0
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0answers
23 views

Confused by Parameterizations and Coordinate Conversions

So I have a few questions regarding parameterizations and coordinate conversion. Ever since dealing with parametric equations last semester I have felt like I have never truly understood ...
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1answer
96 views

How is $\mathbb{C}\times\mathbb{C}$ a real vector space?

I'm working on Linear Algebra homework. I'm having trouble with: $\mathbb{C}\times\mathbb{C}$ is a real vector space. Explain why. Write down a basis for this real vector space. I'm just confused on ...