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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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 $ ...
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1answer
29 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
62 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
47 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
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 ...
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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 ...
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2answers
343 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 manual: ...
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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 $x\...
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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 ...
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1answer
131 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: $$a+(b*...
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1answer
347 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
64 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 Cauchy-...
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3answers
211 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 ...
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2answers
154 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
142 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
36 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 ...
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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 :E\...
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1answer
142 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 ...
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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 ...
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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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45 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
45 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 ...
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1answer
36 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 space,...
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1answer
49 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 ...
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1answer
103 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 ...
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1answer
128 views

Nontrivial subspace of $ \mathbb{{R}^2}$

If $F$ is a non-trivial subspace of $ \mathbb{{R}^2}$,and the vector $v\in F-\left\{ \vec{0} \right\} $ then F= $\left\{ \alpha v;\alpha\in\mathbb{R} \right\} $ .it's easy to show (by the ...
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1answer
41 views

Vectors, dot product

It is given that $|\mathbf{a}|= \sqrt{3}$ and $|\mathbf{b}|= 1$. $\mathbf{a}$ and $\mathbf{b}$ are non-parallel, and the angle between them is $\frac{5\pi}{7}$. I've also found out from the first ...
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2answers
71 views

Do the statements hold in an inner product space over $\mathbb R$ as well?

Let $V$ be an $n$-dimensional inner product space over $\mathbb C$ and $f\in \mathcal L (V)$ normal. Show that: $f^2=f^3 \implies f=f^2 \implies f = f^*$ $f$ nilpotent $\implies f=0$ ...
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2answers
21 views

Relationship between two vectors $\mathbf{a}$ and $\mathbf b$

I am given two vectors, $\mathbf{a}$ and $\mathbf{b}$. Knowing that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors, and $(\mathbf{a \cdot b)b} = \mathbf{a}$, what is the relation between the ...
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1answer
42 views

Vector spaces and infinite cyclic linear transformations II

Let $V$ be the vector space of infinite dimension on the field $\mathbb{Z}_2$. Let's say $$ V=\langle v_1\rangle+\langle v_2\rangle+\dots+ \langle v_n\rangle+\dots$$ where each $v_i$ has order $2$ and ...
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0answers
31 views

Find a normal for an affine hull

How do I find a normal for the affine hull of {[3,1,4], [5,2,6], [2,3,5]}? Thanks
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1answer
33 views

Quick Vector Space Question

I am working on a practice exam and one of the questions is: Always True or False: The set of real numbers $\mathbb{R}$, is a vector subspace of $\mathbb{R}^{1 \times 1}$ I don't really know how to ...
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1answer
41 views

Proving a vector space is, again, a vector space with respect to a new addition-law

Let $(V,+, \cdot)$ be a vector space and $T: V \rightarrow V $ a linear transformation. Let another addition-law in $V$ be defined as $\oplus: v \oplus w = T(v+w) = Tv+Tw$. Prove that $(V, \oplus, \...
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1answer
70 views

Finite dimensional vector space given by polynomials

Let $C[t]$ be the (infinite dimensional) vector space of complex polynomials, and let $0 ≠ g \in C[t]$ be any nonzero complex polynomial. Consider the transformation $ p(f): C[t] \to C[t], f \to g f$....
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0answers
54 views

Vector spaces and infinite cyclic linear transformations

Let $G$ be a direct sum of infinitey many copies of the group of rational numbers $\mathbb{Q}$. Let $\alpha$ be an automorphism of $G$ with infinite order. Is always possible to find an $\alpha$-...
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3answers
156 views

Show $T$ is diagonalizable if $T-\lambda I$ is idempotent

Suppose $V$ is a finite dimensional vector space of dimension $n$ and $T$ is a linear operator on $V$ such that the characteristic polynomial of $T$ splits. Let $\lambda_1,\lambda_2,...,\lambda_k$ be ...
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1answer
83 views

if Dim(U)+Dim(W)>Dim(V) So $U\cap W\neq 0$

Let there be $U,W\subseteq V$ Prove: if $Dim(U)+Dim(W)>Dim(V)$ So $U\cap W\neq 0$ $Dim(U+W)=Dim(U)+dim(W)-Dim(U\cap W)\rightarrow Dim(U\cap W)=Dim(U)+dim(W)-Dim(U+W)<Dim(V)-Dim(U+W)$ Because $...
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0answers
39 views

Transform to cylindrical coordinate system

I tried so many approaches , at least give me a hint on how to find The unit vectors $$ \vec{V} = y\vec{i} + x\vec{j} + \frac{x^2}{\sqrt[2]{x^2+y^2})} \vec{k}\ $$
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0answers
34 views

How to show the planet always stays in the same plane?

I know angular momentum $q \times p$ is conserved, where $p=L_{\dot q}$ is linear momentum. How to apply this to a planet orbiting the star, described by the position vector $q$ relative to the star. ...
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1answer
47 views

Is there a difference between “in the direction of (1.1)” and “in the direction toward (1.1)?”

The question that I have is A differentiable scalar field f has, at the point (1.2), directional derivatives 2 in the direction toward (2.2) and -2 in the direction toward (1.1) Determine the ...
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1answer
86 views

Vector space example

What is an example of a vector space that is a non-linear map from the real vector space of all real-valued continuous functions on R to itself?
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1answer
264 views

Matrix Linear Transformations in R3

I find this to be a very interesting problem. I extracted the vectors into a[1 5 -3] and b[2 -1 4]. For part (a), I know that the subspace is simply a space within the space R3. How would one go about ...
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2answers
55 views

Finding Range of Transformation of convergent sequences

$V$ is a vector space of all real convergent sequences. Define a transformation $T : V \rightarrow V$ s.t. if $x = \{x_n\}$ is a convergent sequence with limit $a$, $T(x) = \{y_n\}$, where $y_n = a - ...
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2answers
69 views

Prove that $\{v_1,v_2…v_n\}$ is linearly dependent if $a\notin span\{v_1,v_2,v_3…v_n\}$ and $v_n\in span\{v_1,v_2…v_{n-1},a\}$.

I ran into the next problem and got really confused: Let $\{ v_1, v_2,v_3... v_n \}$ be a set of vectors in the vector space $V$, and let $a\in V$ in such a way that: $a\notin span\{v_1,v_2,...
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1answer
28 views

Dimesion of a subspace subject to linear constraints

Suppose $X$ is $n\times K$ with full column rank $K$ and $G$ is $q\times K$ with full row rank $q$. If $q<K$, how do I see that $\mathcal{L}\equiv\{Xb,b\in\mathbb{R}^K,Gb=0\}$ has dimension $K-q$...
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0answers
29 views

Is the following a subspace, where $f(t) \in C^{1}[0,1]$?

This is probably an easy question, but I just want to check my understanding of it. Here it is: Let $f(t) \in C^{1}[0,1]$. Is the subset of functions $f(t)$ such that $\frac{df}{dt}+\int_{0}^{1}f(...
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2answers
197 views

How do I verify that a set of vectors is a basis for the given plane

I have a set of 2 vectors: $\{ (1,2,0), (0,2,-1) \}$. I have to show that this set is a basis for the plane with equation: $2x_1 - x_2 -2x_3 = 0$. I know that the normal vector of the plane is $\...