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
votes
0answers
20 views

How to understand the geometry of bilinear forms that are not positive-definite?

I simply cannot find a good resource that explains intuitively how to understand the geometry that is induced on a vector space when the bilinear form is not positive-definite. In the ordinary ...
-2
votes
0answers
29 views

What are some unanswered problems on vector algebra? [on hold]

I'm sorry for any mistakes.Math language is different from my country's so they may be so wrong tags. Thanks in advanced ^^
0
votes
1answer
32 views

A program to visualize Linear Algebra?

I am asking here because I believe you have some idea of a good visualizer 3d program to see what are really: eigenvectors, subspaces, rowspaces, columnspaces and just answers on normal matrix ...
3
votes
0answers
50 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
1
vote
1answer
34 views

Confusion regarding dimension of a vector space

The dimension of a vector space is the number of elements in the basis for that vector space. If we look at $\mathbb R^n$, then we say that the dimension of $\mathbb R^n$ is $n$. So every element in ...
0
votes
0answers
11 views

Equivalent definitons of Centerpoint

In Jiri Matousek's book - "Lectures on Discrete Geometry", he defines centerpoint as: 1.4.1 Definition (Centerpoint). Let $X$ be an $n$-point set in $\mathbb R^d$. A point $x \in \mathbb R^d$ is ...
0
votes
1answer
18 views

Find vector components given vector modulus

Given the modulus $v$ of a vector $\mathbf{\bar{v}}$, is there a way to find any possible single combination of the vector components $v_x,v_y,v_z$ such that $$v^2 = v_x^2 +v_y^2 +v_z^2 $$ I am ...
0
votes
2answers
24 views

Simple exercise regarding space spanned by two vectors

Consider the vectors $X_1=(1,3,2)$ and $X_2=(-2,4,3)$ in $\mathbb R^3$. Show that the set spanned by $X_1$ and $X_2$ is given by $\{(Y_1,Y_2,Y_3):Y_1-7Y_2+10Y_3=0\}$.
0
votes
1answer
20 views

What does it mean for a polynomial to be positive semi-definite?

I'm reading proofs about Cauchy-Schwarz inequality. Some of them use the argument of a polynomial being positive semi-definite as the starting argument. That is, they argue that $$\forall x,y ...
0
votes
1answer
20 views

Gradient of Norm 2 with chain rule

Assume $A$ is a matrix with size of $m*n$ and $b$ is a vector with size of m. If $f$ is a function which accepts a scalar and returns a vector with size of $n$. Now what is the gradient of following ...
-5
votes
0answers
31 views

Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
1
vote
0answers
36 views

$\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$

Let $A:\mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. We know that there is a unique transformation $A^*:\mathbb{R}^n \to \mathbb{R}^m$ such that $$\langle Ax,y\rangle = \langle x,A^*y ...
0
votes
1answer
13 views

cardinality of fiber in a finite morphism of schemes

Given $f:X\to Y$ a finite morphism of schemes, with $Y$ locally noetherian, let's take a point $q\in Y$, and an affine noetherian open set $$q\in U=Spec(B)\subseteq Y$$ Then ...
0
votes
0answers
19 views

Solving a problem that is function of vector (norm 1)

Assume we have a vector ${\bf v}$ of complex entries and we are trying to solve for the following $$ \text{Re} ({\bf v} ^H e^{ j \text{angle} ({\bf v})})=?$$ where $\text{Re}$ means the real part of ...
0
votes
1answer
22 views

How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
0
votes
2answers
12 views

Given a vector and angle find new vector

Given a vector and an angle, how can i find an vector that the angle between the two vector is exactly the given angle? Edit: We are in the n-dimensional space and the new vector has a fixed given ...
0
votes
0answers
20 views

How to find Tangent Normal Binornmal vectors of parametric knot

I am given parametric equations of torus knot: $$x = (a+b\cos(qt))\cos(pt)$$ $$y= (a+b\cos(qt))\sin(pt)$$ $$z= c\sin(qt)$$ where $0<t<2\pi$. I need to find Tangent, Normal and Binormal ...
0
votes
2answers
33 views

Looking for the basis of the kernel of T

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
1
vote
0answers
33 views
1
vote
1answer
28 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
0
votes
1answer
40 views

What is a change of basis and how do i find it?

W is a four dimensional vector space over a field F with basis S = (v1, v2, v3, v4). B is also a basis of W such that. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...
6
votes
1answer
21 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
3
votes
2answers
55 views

Does the set of all piecewise constant functions form a subspace of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$?

A function $f\in \mathbb{R}^\mathbb{R}$ is piecewise constant if and only if it is a constant function $x\to c$ or there exist $a_1<a_2<\cdots<a_n$ and $c_0,...,c_n$ in $\mathbb{R}$ such that ...
1
vote
1answer
32 views

The weak topology on an infinite dimensional linear space is not first-countable

I thought I needed help proving the above statement, but during typing I found a proof. Since I had already written it all down I will post it anyway, maybe in the future someone can benefit from it. ...
2
votes
1answer
27 views

Dimension of a single coordinate point in $\mathbb{R}^2$?

While going through the comments on an interesting topic on MathOverflow, I came a cross a quote: Take three distinct lines in R^2 as U, V, W. All intersections have 0 dimensions. I have only ...
0
votes
1answer
42 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
1
vote
1answer
49 views

How to show T $P_2 \, \to \, P_2$ is a linear operator. Find a basis for the kernel T?

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
0
votes
0answers
19 views

> Find the matrix A for which $[T(p(x))]_B$= for all p(x) $\in$ P2

Hey i'm quite confused with this question please link me so i can understand the theory. The question is. Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree ...
0
votes
0answers
24 views

If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
0
votes
2answers
19 views

A set of vectors forming the basis

Can: (b) A set of four vectors: {(1,2,3),(2,3,1),(3,1,2),(1,3,2)} form a basis for $\mathbb{R}^3$?
2
votes
2answers
35 views

Is $V = \{(x,y,z)\in \mathbb{R}^3:\ x+y >1 \}$ a subspace?

Prove whether the following subsets of $\mathbb{R}^3$ are subspaces : (a) $$V = \{(x,y,z)\ \in \mathbb{R}^3:\ x+y >1 \ \},$$ I think that this is not a subspace as the zero vector does not ...
2
votes
1answer
17 views

Finding A Spanning Set

How do I find the spanning set for: $$V = \{(2a,b,0)\ :\ a,b \in \mathbb{R} \},$$ where $V$ is a subspace of $\mathbb{R}^3$?
5
votes
0answers
59 views

Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
1
vote
2answers
15 views

What am I doing wrong in finding the orthogonal projection of a vector onto the subspace V?

Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$. Find the orthogonal projection of $w$ onto V,using exact values in your answer. My Approach Let the ...
0
votes
0answers
31 views

Largest algebra in a vector space

Let $V$ be a vector space and let $C$ be a subset of $V$ that is closed under a bilinear operator $\langle \cdot, \cdot \rangle: V \times V \rightarrow V$. Let $A \subset V$ be an algebra containing ...
1
vote
1answer
14 views

Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
0
votes
0answers
12 views

Given the angles of a 3d vector and the length of one of the components find the length of the other two components

The angles of a vector are 118 with the positive x axis, 76 with the positive y axis and 148 with the positive z axis. The y direction component of the vector is 5. How do you find the other two ...
1
vote
0answers
29 views

Checking if this set is a vector sp.

Question: Define a set $V=\{(x,y):x,y\in\Bbb R\}$. For any two elements $u=(u_1,u_2),v=(v_1,v_2)$ in $V$ and $t\in\Bbb R$, addition and scalar multiplication as, ...
1
vote
1answer
33 views

How are vector space dimension and basis related?

How are vector space dimension and basis related? (I am new to these concepts and know little to nothing about linear algebra/advanced calculus.) Thank you in advance.
0
votes
1answer
15 views

Given a set of complex subspaces, find a set of disjoint subspaces such that every original subspace is the span of the union of some subset

Suppose $S$ is a set of subspaces of $\mathbb{C}^{n}$ for some integer $n$. I would like to find a set $T$ of disjoint subspaces (not just pairwise disjoint - is there a clearer word for this?), such ...
4
votes
1answer
279 views

How can this be a vector space?

I found the following statement: "Example of a linear - vector - space: The set $C^{(k)}[a,b]$ of all (real-valued) continuous functions on a finite interval $a ≤ t ≤ b$ with addition and real number ...
-1
votes
5answers
32 views

Find two vectors v1 and v2 such that when added equal (0, 4, 0).

Struggling with this question. Find two vectors $v_1$ and $v_2$ such that when added equal $(0, 4, 0)$. $v_1$ is parallel to $u(-2, 4, -2)$ and $v_2$ is perpendicular to $u$. Not sure how to start.
2
votes
3answers
52 views

Eigenvalues of matrix with all $1$'s. [closed]

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
1
vote
0answers
33 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
0
votes
0answers
33 views

Row rank$=$Column rank

This is one of the proofs given on Wikipedia. Let $A$ be an $m \times n$ matrix with entries in the real numbers whose row rank is $r$. Therefore, the dimension of the row space of $A$ is $r$. Let ...
0
votes
0answers
27 views

Looking for a vector space $V$ and $T \in \mathcal L$ such that $ker (T) \cap Im(T)=\{\theta\}$ but $V \ne ker(T)+Im(T)$

I am looking for example of a linear operator $T$ on a vector space $V$ such that $ker (T) \cap Im(T)=\{\theta\}$ but $V \ne ker(T)+Im(T)$ . I know that $V$ cannot be finite dimensional . Please help ...
0
votes
1answer
23 views

to find basis of homomorphism

Compute $Hom(V,W)$ and also determine its dimension over $F$ where $V$ and $W$ are vector spaces over the Field $F$ given that $V=\mathbb R^2, W=\mathbb R^2, F=\mathbb R$ I have done this: ...
23
votes
3answers
531 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...
0
votes
1answer
31 views

Subspaces of an infinite dimensional vector space

It is well known that all the subspaces of a finite dimensional vector space are finite dimensional. But it is not true in the case of infinite dimensional vector spaces. For example in the vector ...
0
votes
2answers
30 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...