For questions about vector spaces and their properties. More general questions about linear algebra belong under the 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 where we ...
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2answers
25 views
Boundedness of Surfaces in $\mathbb R^3$
GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?
3
votes
1answer
298 views
For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
3
votes
1answer
187 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 ...
3
votes
2answers
100 views
Why is the maximal value attained at the boundary?
Let $A$ be a real matrix.
Denote $\|\cdot \|$ the $p=1$ norm (sum of absolutes of the elements).
Let $C$ be all vectors (of compatible size with $A$) whose elements are in the range $[-1,1]$
How to ...
3
votes
1answer
2k views
How do I determine if the vectors lie on a plane in an $N\times N$ matrix?
In a $2\times2$ matrix, it is quite easy to see if the vectors lie on a plane or not. By vector, I mean the columns of the matrix. I usually determine if the numbers are of a certain multiple. From ...
3
votes
3answers
290 views
Proving these basic properties of subspaces of vector spaces
I come across an interesting problem on my journey of cracking open some old math books and cracking down on problems from boredom. I cannot seem to wrap my head around this problem of subspaces. The ...
3
votes
1answer
411 views
A question on finding the intersecting line between two planes
According to my math book, in order to find the intersecting line between two planes we need to:
Find the vector product of the direction normals of the two planes
Write the equations of the planes ...
3
votes
1answer
48 views
Does $n^A\cong n^B \Rightarrow A\approx B$ require choice?
Let $V$ be a vector space over a finite field $F$.
Assume $V$ has a basis $S$.
Then, define $\Phi:V^*\rightarrow F^S:f\mapsto f\upharpoonright S$. It can be shown that $\Phi$ is an isomorphism. Thus, ...
3
votes
2answers
90 views
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
3
votes
1answer
61 views
is this solution of $AX=0$ in space VA?
Asume in $\mathbb{R}^4$ $$AX=0 \,(\text{mod } 11)$$ with
$$A=\begin{bmatrix}1 & 2 & 3 & 4\\ 1 & 2 & 3 & 4\\1 & 2 & 3 & 4\\1 & 2 & 3 & 4\end{bmatrix}$$
...
3
votes
1answer
88 views
Looking for proof that an open set in vector space contains the sum of two open sets.
Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
3
votes
1answer
125 views
What is the name of this equation?
I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture?
I have also tried to re copy it:
$$
...
3
votes
1answer
277 views
When are two diagonal matrices congruent?
This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
3
votes
2answers
162 views
Vector Force Fields and Their Physical Interpretations
The vector force field F=(yi,-xj) has a curl of -2. The acceleration of a particle in space is given by:
ax=y/m
ay=-x/m
This vector field has a divergence of 0. Will particles in this vector FORCE ...
3
votes
2answers
54 views
Proof of the linear independence of the generalized eigenvectors of a square matrix
I'm currently stuck on this problem:
Let $V$ be a finite dimensional vector space. If $S: V\rightarrow V$ and $T: V\rightarrow V$ are linear maps and $ST=TS$, prove every eigenvalue of $ST$ is a ...
3
votes
1answer
46 views
Eigenvalues and Eigenvectors Diagonilization
Let $ A=\begin{bmatrix}
-7 & -1 \\
12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$.
Ok so the first thing I need to look ...
3
votes
1answer
57 views
Some questions about quaternions.
It is possible make something like complexification of a real vector space using quaternions?
If yes, it's similar to complex case or there are considerable differences?
Has been studied a quaternion ...
3
votes
1answer
65 views
Does anyone know any resources for Quaternions for truly understanding them?
I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
3
votes
1answer
312 views
How to show the unit ball of the dual norm is also polytope?
Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
3
votes
2answers
111 views
Matrix proof using norms
I have a linear algebra question I need help with.
Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
3
votes
1answer
89 views
Finding the equation of a line entirely defined by a three variable equation.
How can you find the equation of a line that lies completely in a set defined by a three variable equation. For instance, the equation of a line entirely in the set defined by $x^2 + y^2 - z^2 = 1$
...
3
votes
1answer
96 views
Grassman formula for vector space dimensions
If $U$ and $W$ are subspaces of a finite dimensional vector space, $$ \dim U + \dim W = \dim(U\cap W) + \dim(U + W)$$
Proof: let $B_{U\cap W} = \{v_1,\ldots,v_m\}$ be a base of $U\cap W$. If we ...
3
votes
1answer
83 views
Are the constant functions a closed subspace in the polynomials?
Consider all polynomials $\mathbb R[x]$ and the subspace of polynomials of degree $0$, which we will refer to by the letter $U$. Is this subspace closed with respect to the inner product: $$\langle ...
3
votes
1answer
58 views
Can one always make hybrid bases from two bases?
Suppose you have a finite dimensional vector space $V$ with two different bases $\{b_1,\dots,b_n\}$ and $\{c_1,\dots,c_n\}$. Is it possible to swap out any number of vectors between the two bases such ...
3
votes
2answers
120 views
Finding $\text{ker}(Tr)$
Defining the trace in the usual way as a function $Tr: F^{n\times n} \rightarrow F$, where $F$ is some field. I want to show that $\text{ker}(Tr)=\text{span}_F(\{AB-BA|A,B\in F^{n\times n}\})$.
So ...
3
votes
1answer
150 views
Splitting Exact Sequences
Given an exact sequence of vector spaces: $$0\longrightarrow U \longrightarrow V \longrightarrow W\longrightarrow 0$$ with $f:U\rightarrow V$ and $g: V \rightarrow W$
I want to prove the that ...
3
votes
1answer
57 views
Finding some isomorphisms
Letting $U, V$ be vector spaces over $\mathbb{F}$ with $W\subseteq V$ a subspace. I want to show that if $B = \{T\in Hom_\mathbb{F}(U,V) | im(T)\subseteq W\}$ that $$B\approx ...
3
votes
3answers
364 views
Question on finite Vector Spaces, injective, surjective and if $V$ is not finite
Let $V$ be a vector space and $\alpha \in \operatorname{End}(V)$
(i) If $V$ is finite dimensional, then $\alpha$ is injective iff $\alpha$ is surjective.
(ii) Give example showing (i) is false if ...
3
votes
1answer
58 views
Is the vector in the space of 3 other vectors
I have a set of 3 vectors
$$ IE = {[1, 1, -3]; [2, -1, 3]; [-6, 3, -9]}$$
I want to know if the vector [1, 4, -12] , belongs (or is in the span?) to my previous set.
So here's what I did.
$$
...
3
votes
1answer
133 views
Is the set of all power series with convergence radius equal to $1$ a vector space?
Given the set of all power series with radius of convergence ($r$ in the definition) equal to one:
$$A:=\{\sum a_kz^k | r =1\}$$
Does $A$ form a vector space?
The radius of convergence doesn't ...
3
votes
3answers
271 views
Predicting the next vector given a known sequence
I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a ...
3
votes
1answer
147 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 ...
3
votes
2answers
130 views
Nonsingular bilinear map
This is related to a previous question I asked. But I realize that my logic there is total bonkers. And it would be great if someone could help me out a bit.
$B:V\times V\to F$ is a bilinear form ...
3
votes
1answer
110 views
Subspace generated by permutations of a vector in a vector space
Let $K$ be a field. Consider the vector space $K^n$ over the field $K$.
Suppose $(a_1,a_2, ... ,a_n) \in K^n$. What is the dimension of the subspace generated by all the permutations of ...
3
votes
1answer
86 views
What are some nice examples to illustrate that a basis for $V^\ast$ induces a basis for $V$
Let $V$ be a $n$-dimensional vector space over $\mathbf{C}$ and let $(v_1^\ast,\ldots,v_n^\ast)$ be a basis for $V^\ast$. Then, there is a unique basis $(v_1,\ldots,v_n)$ for $V$ such that ...
3
votes
2answers
78 views
Dimensions and inequalities
How does one go about proving that $\dim(T(X))\geq \dim(T(V)) - \dim(V) +\dim(X)$ where $X$ is a subspace of $V$, a vector space, and $T$ is a linear transformation? Thanks.
3
votes
2answers
225 views
How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$?
Let $x, y$ be two complex vectors, $$\cos\Theta(x,y):=\operatorname{Re} \frac{y^*x}{\|x\|\|y\|} .$$ Then I want to prove that $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$
3
votes
1answer
60 views
Can a closed (non-trivial) subspace of an incomplete vector space be complete?
While thinking about the statement:
A subspace of a complete vector space is closed if and only if it's complete.
I was trying to drop the first "complete" and see what gets broken.
And my ...
3
votes
1answer
71 views
$AX=C$: An Inconsistent Linear Equation [duplicate]
Question:
Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C \in F^n$ such that
the system of linear ...
3
votes
1answer
76 views
Finding intersection of 2 planes without cartesian equations?
The planes $\pi_1$ and $\pi_2$ have vector equations:
$$\pi_1: r=\lambda_1(i+j-k)+\mu_1(2i-j+k)$$
$$\pi_2: r=\lambda_2(i+2j+k)+\mu_2(3i+j-k)$$
$i.$ The line $l$ passes through the point with ...
3
votes
1answer
96 views
Let $n$ be an integer $\geq 2$ and let $M_n(\Bbb R)$ denote…
Let $n$ be an integer $\geq 2$ and let $M_n(\Bbb R)$ denote the vector space of $n \times n$ real matrices. Let $B \in M_n(\Bbb R)$be an orthogonal matrix and let $B^t$ denote the transpose of $B.$ ...
3
votes
1answer
63 views
$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?
Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
3
votes
1answer
38 views
bound on $l_2$ error in approximating a vector with its $t$-sparse representation
How do I prove that for any vector $y\in \mathbb{R}^n$, and any positive integer $t$,
\begin{equation}
||y-y_t||_2\:\leq\: \frac{1}{2\sqrt{t}}||y||_1
\end{equation}
where $y_t\in\mathbb{R}^n$ is the ...
3
votes
1answer
130 views
Finding the dimensions of subspaces of a Vector space and S-cyclic subspaces using minimal poynomials
I've been staring at a chapter in Bill Cooperstein's Advanced Linear ALgebra for some time now and one section is giving me trouble. It is about elementary divisors and invariant factors. My ...
3
votes
1answer
78 views
The multiplication of 2D vectors produces what?
I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication.
To avoid confusion with other types of multiplication, this is the basic form I ...
3
votes
1answer
96 views
what's the intuition of the transpose of a matrix
I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
3
votes
2answers
53 views
Symmetry in ordinary differential equations
Suppose I am given an ode $${dy\over dx}={1\over x^2}f(xy)$$ where $f$ is some arbitrary function.
How then does doing the following help solve the equation? :
First I have a vector field ...
3
votes
1answer
115 views
“Algorithmic” proofs in linear algebra
Although I am new to linear algebra, I want to study it with as much rigor as possible. After searching around, I picked up Halmos' Finite Dimensional Vector Spaces and Axler's Linear Algebra Done ...
3
votes
1answer
89 views
Linear Transformation over Subfield
Letting $F\subseteq K$ be fields, and $V$ a vector space over $K$. Certainly, $V$ is also a vector space over $F$. And if $\{e_1,...,e_n\}$ is a basis for $K$ over $F$ and ...
3
votes
1answer
120 views
Proving something is the basis of a quotient space
Let $k$ be a field which does not have characteristic 2. Let $M$ be the free $k$-vector space generated by two elements $\{ c, x \}$. Let $T(M)$ be the tensor algebra of $M$ and let $I$ be the ideal ...
