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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1answer
153 views

Showing unique decomposition into parallel and orthogonal parts for any subspace

Given a general (possibly infinite dimensional) linear vector space $V$ with an inner product, how can you prove that for any subspace $S$, any vector v in $V$ can be uniquely expressed as $$v = s + ...
0
votes
1answer
56 views

If $F$ and $R$ are subspaces of vector space $E$, then $F \cap R \neq \varnothing$

I need to prove the following: let $F \cap R$ intersection of vector subspaces $F$ and $R$ of vector space $E$, then $F \cap R \neq \emptyset$ Thanks in advance!
0
votes
2answers
723 views

Non-parallel vectors confusion

I've got a section in my textbook about non-parallel vectors, it says: For two non-parallel vectors a and b, if $\lambda a + \mu b = \alpha a + \beta b$ then $\lambda = \alpha $ and $\mu = \beta $ ...
0
votes
3answers
104 views

Diagonalizable Operators: An Operational Extension

Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator $$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$ on $V$ is also diagonalizable for any scalars $a_1, ...
1
vote
0answers
49 views

Hahn Banach to get linear functional bounded by sub/superlinear functionals

I am working in a real vector space $V$. I have seen it written that if I have a sublinear functional $p$ and a superlinear functional $q$ such that $q \le p$ then there exists some linear functional ...
2
votes
1answer
87 views

Basic concepts in finite fields

I need some help with clearing up some some basic concepts in finite fields. I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to ...
1
vote
1answer
43 views

The algebraic possibilities of the (topological) procedure of the compactification of a space

If $X$ is locally compact $K$-vector space, then $X\cup \{\infty\}$ is via the Alexandroff-compactification a compact space. But this purely topological procedure tells me nothing about the algebraic ...
0
votes
2answers
66 views

Ray-Lens Intersection

So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector. I also have a spherical lens with aperture ...
0
votes
0answers
90 views

Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin

I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
1
vote
3answers
1k views

real numbers a vector space over rational numbers? [duplicate]

Let $V$ be set of real numbers and $K$ the field of rational numbers. Is $V$ a vector space over $K$, with ordinary addition of real numbers and multiplication by rational numbers?
1
vote
1answer
54 views

What does this mean: Symmetry of the KDV generated by a vector field

What is a symmetry of the KDV $$\frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^3 u}{\partial x^3}$$ generated by $$V=A(t,x,u)\frac{\partial }{\partial ...
2
votes
1answer
100 views

Given a vector space $V$, show that the following statements are equivalent.

Given a subset $W$ of $V$ then I want show that, $\forall v \in V, w \in W$ $\exists \lambda \in \mathbb{R}$ such that $w + \alpha v \in W$ for any $0 < \alpha < \lambda$ iff $\forall v \in ...
0
votes
3answers
2k views

Vectors that form a triangle!

I have a problem here. How can I prove that sum of vectors that form a triangle is equal to 0 $(\vec {AB}+\vec {BC}+\vec {CA}=\vec 0)$ ? Thank you!
0
votes
3answers
140 views

$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
0
votes
2answers
119 views

Linear Algebra - Vector Spaces of Vector Spaces

If we have a Vector Space such as $\Bbb R$, we can make Vector Spaces out of it. For example, let $v_1,v_2,v_3\in \Bbb R$. We know that $(v_1,v_2,v_3)$ is a Vector Space - it is $\Bbb R_3$ - and its ...
0
votes
2answers
188 views

How to show that a equation is a vector space?

Let $W$ be the set $W := \{(p,q,r,s) \in \mathbb{R}^{4} \mid x+3y+4q=0 \}$. How can I show that $W$ is a vector space? Is $(6,8,6,4)$ in $W$? How and why? Please do all the works I am really confused ...
0
votes
2answers
86 views

Which of the following are subspaces of $M$?

Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$ Which of the followings are subspaces of $M?$ ...
3
votes
0answers
46 views

find the dimension of $W.$

Let $W=\{p(B):p \text{ is a polynomial with real coefficients}\},$ where $B=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}.$ Then find the dimension of $W.$ I have shown ...
0
votes
2answers
122 views

Is $w$ a vector space?

Let $W$ be the set of all solutions$(X,y,i,r)$ such that $a+b=m^2$. is $w$ a vector space? Can anybody do the whole thing for me and explain shortly every step? I have to do this kind of lots of ...
1
vote
1answer
45 views

Find vectors vertical to given vectors with certain length

Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1 Every vector $\mathbf{x'}$ that is to be found must meet these ...
0
votes
1answer
929 views

Vector Field, Scalar Field. Which is meaningful and not?

Let a, b, c be vectors, f(x, y, z) be a scalar field, F(x,y,z) be a vector field. Which of the following expressions are meaningful? I. (a×b)×(c×b) II. |a|(b· c) +|a|(b+c) III. ∇ ×(f F) IV. (∇ ...
2
votes
1answer
1k views

Relationship between covariant/contravariant basis vectors

I'm starting to learn some of the basics of covariant and contravariant vectors. I'm a little confused about the difference between a covariant and a contravariant basis vector. I know that the ...
0
votes
3answers
90 views

When are two vectors parallel if the vectors are $5e_1-3e_2+\alpha e_3$ and $\beta e_1 + 2e_2 + 3e_3$

When are two vectors parallel if the vectors are $$5e_1-3e_2+\alpha e_3$$ and $$\beta e_1 + 2e_2 + 3e_3$$ The alpha and beta are parameters.
1
vote
0answers
53 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
0
votes
1answer
51 views

Bilinear Forms: An Initial Condition Proof

Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
2
votes
3answers
170 views

Calculate two vectors given their norms and angle

For two vectors $\mathbf{u,v}$ in $\mathbb{R}^n$ euclidean space, given: $\|\mathbf{u}\| = 3$ $\|\mathbf{v}\| = 5$ $\angle (\mathbf{u,v})=\frac{2\pi}{3}$ Calculate the length of ...
0
votes
1answer
50 views

Linear interpolation of points in isometric isomorphic spaces

Suppose that we have two spaces $\mathcal{F}$ and $\mathcal{H}$ and we know that $\mathcal{H}$ is isometric isomorphic to $\mathcal{F}$, so that distances and angles are preserved. Note that we are ...
2
votes
3answers
74 views

How to decompose a vector regarding complementary subspaces

How to show that $U_1:=\langle(1,1,1)^T\rangle$ and $U_2:=\langle(1,0,0)^T,(0,0,1)^T\rangle$ are complementary subspaces and how to decompose $(1,2,3)^T$ regarding these ? Best Regards, Thomas
3
votes
1answer
144 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
1k views

Vector 2 norm and infinity norm proof

So I've already proven why $\left\lVert x\right\rVert_2\geq \left\lVert x\right\rVert_\infty$. I'm having trouble proving that $\sqrt{m}{\left\lVert x\right\rVert_\infty}\geq \left\lVert ...
1
vote
2answers
70 views

Prove using an example that there is no plane on $\mathbb{R}^3$ that contains every group of 4 points

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere): The exercise is as follows: i) Find the equation of the plane of $\mathbb{R}^3$ ...
16
votes
1answer
225 views

Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
3
votes
1answer
99 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 ...
1
vote
2answers
129 views

Inconsistent System of Linear Equations

Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$ has more than one solution. Prove that there is a column $C ∈ F^n$ such that the system of linear equations $AX = C$ is ...
1
vote
2answers
75 views

Newbie vector spaces question

So browsing the tasks our prof gave us to test our skills before the June finals, I've encountered something like this: "Prove that the kernel and image are subspaces of the space V: $\ker(f) < V, ...
1
vote
3answers
51 views

Reordering vector product

If I have vectors $a, b, c \in \mathbb{R}^3$, and if we have e.g. $a = b\times c$, is there any way to express $b$ in terms of the other two?
1
vote
1answer
439 views

Infinitely many solutions vs one solution vs no solution in systems involving an unknown constant

Just need a little clarification in case my assumptions are incorrect. If I were to have the matrix {{1, 3, 3}, {2, 7, 6}, {1, 4, k^2 - 18}} * {x1, x2, x3} = {1, 3, k+1} which has a row echelon ...
2
votes
1answer
40 views

Range of adoint operator

We consider infinite dimension. $X,Y$: Banach Spaces $T:X→Y$ is a bounded linear operator. I want to prove $(\ker\, T)^\bot = \overline {R(T^*)}$. $(\ker\, T)^\bot = \{f\in X^*|f(x)=0\ (x\in ...
-1
votes
2answers
47 views

vector matrices

Given that $u=(1, 1, 1)$, $v = (-2, 1, 0)$ and $w = (0, 3, 0)$, calculate the following (where possible): ...
3
votes
2answers
161 views

Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$

So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
0
votes
1answer
30 views

Vector direction

I have a vector in the 2nd Coordinate of the Cartesian plane. I want to know that how can I find out the direction of the vector that whether it is towards the ...
0
votes
1answer
72 views

I need a proof for a scalar product - no numbers allowed

Hello I simply cant explain to myself why this equation holds. Lets say we have an orthonormal basis $\vec{i}, \vec{j}$ and 2 2-D vectors in this basis which are: \begin{align} \vec{a} &= ...
0
votes
2answers
590 views

Calculate distance from plane to parallel plane in O using vector and normal

I'm trying to figure out what's the best method to get the distance between two planes where i have the normalized vector of the plane and a point in the plane. What I want to do is to create a ...
0
votes
1answer
200 views

least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
5
votes
0answers
188 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
1
vote
0answers
112 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
0
votes
2answers
50 views

Regarding vector spaces

If A and B are n dimensional vector spaces 1) Is A+B a vector space? 2) Is A and B a vector space?
0
votes
2answers
90 views

$X∈M_{m×1}(F)$ and $Y∈M_{1×n}(F)$: A Range Dimension Implication

Let $A \in M_{m \times n}(F)$. To prove that $\operatorname{rank}(A) \le 1$ if and only if there exist $X \in M_{m \times 1}(F)$ and $Y \in M_{1 \times n}(F)$ such that $A=XY$ where must I start?
1
vote
1answer
48 views

$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$

Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
1
vote
1answer
995 views

How to find the gradient for a given discrete 3D mesh?

I have a 3D mesh that is looking like this: ie I have a set of triangles in a 3D space, and they are all linked by their edge. I have to compute the gradient associated with this field, at each edge ...