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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1answer
9 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
-2
votes
0answers
12 views

Find the dimensions spanned by the vectors [on hold]

Compute the dimension of the subspace spanned by each subset. $\{1, e^{ax}, xe^{ax}\}, \{1, \cos 2x,\sin^2 x\}$
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0answers
19 views

Defining a function on a Euclidean Vector Space

Let $\{x, y\}$ be a linearly independent set in a Euclidean space V. Define $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(a) = \|x − ay\|$ I do not understand the question. How should I define that ...
0
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1answer
24 views

The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
0
votes
1answer
14 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
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1answer
27 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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votes
2answers
21 views

Direct Sum of $n$ Subspaces

I just need some guidance to prove a portion of the following theorem. Let $V_1, V_2, ... , V_n$ be subspaces of a vector space $V$. Then the following statements are equivalent. $W = \sum ...
1
vote
1answer
32 views

Find bases for kernel and image of T where $T: P_2 \to M_2$

T is defined as $$T:P_2(\mathbb R) \to M_2 (\mathbb R) \ \text{where} \ T(ax^2 +bx+c)=\begin{pmatrix}-2a +c & b+c\\-3b-3c&6a-3c\\ \end{pmatrix}$$ and I need t find bases for $ Im(T) $ and ...
0
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1answer
14 views

Find the basis when integration is in the condition

Let $V$ be the set of all polynomial $f(x)$ in $P_2$ s.t. $\int_{0}^3 f(x) dx =3f(1) $ If $V$ is a subspace of $P_2$ find a basis of $V$. Can somebody help me get started? The integral condition ...
2
votes
2answers
54 views

Finding whether a vector is in the span of a set of vectors

$$U=\operatorname{span}(v_1,v_2,v_3)$$ where \begin{aligned} v_1=(1,1,1,2) \\ v_2=(1,2,3,1) \\ v_3=(0,1,2,-1)\end{aligned} I need to find if $u=(2,1,0,5) \in U$ If $u \in U$ it is a linear ...
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2answers
17 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
0
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2answers
236 views

How to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$ based on the following assumption?

Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$? My approach is to use ...
-1
votes
1answer
38 views

isomorphic linear spaces [on hold]

Let $S$ be the space of $3\times 3$ skew-symmetric real matrices. Then $dim\ S=3$. Is it true that $S$ is a vector isomorphic to $\mathbb R^3$? What is the isomorphism then?
5
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1answer
21 views

Zauner's conjecture

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 ...
0
votes
1answer
43 views

Vectors spaces: $V=U+W$. Technique for showing that an element belongs to $U$?

Let $V$ be finite-dimensional vector space and let $U$, $W$ be subspaces of $V$. Suppose that $V=U+W$. Is there a standard argument for showing that an element belongs to $U$? For example, if the ...
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votes
2answers
48 views

Prove that color space a vector space [on hold]

I am trouble in proving that color space is a vector space. Please if anyone knows how to prove this, share your solution. Just consider RGB color model as it can be represented as 3 vectors. Now we ...
2
votes
1answer
49 views

Find the kernel of the linear transformation

So the question asks: find the kernel of the linear transformation $T : \mathbb{R}^4 \to \mathbb{R}^3$ defined by $T(x) = Ax$ where $A$ is the matrix: $$\begin{bmatrix}1 & 0 &1 & 0\\0 ...
0
votes
3answers
33 views

Suppose $U$ and $W$ are subspaces of the vector space $V$. Show that $U + W$ = $sp(U \cup W)$

I am not sure where to start with this one, any help would be appreciated. I tried using the definition of a span but I couldn't see where to go from there.
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votes
1answer
17 views

Complementary subspace of $M=(p(2x)=p(x)) ,p\in P_4$

Can anyone please help me with: Find a some base for complementary subspace of $$M=(p\in P_4 : p(2x)=p(x+1)), $$
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2answers
43 views

Prove that $V^ \bot$ is a subspace of $R^n$ prove

Let $V$ be a subspace of $R^n$. Let $V^ \bot$ be a subset of $R^n$ defined by: $V ^ \bot$ = {$\vec x \in R^n$: $\vec x * \vec v = 0$ for all $\vec v \in V$} Prove that $V^ \bot$ is a subspace of ...
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2answers
36 views

Why are these not bases in $\mathbb{R}^4$?

I know that bases vectors must span and be linearly independent. The (i) is not bases because the last vector contains $\pi$. The (iii) is not bases because they are not linearly independent. The ...
0
votes
1answer
21 views

Why is a map to a smaller dimensional space not injective?

I am trying to proof the theorem that states that: "A map to a smaller dimensional space is not injective" So I first suppose that V and W are finite-dimensional vector spaces such that dimV > dimW ...
1
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4answers
39 views

Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$

Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$. I know how to ...
0
votes
1answer
52 views

subset $W$ of $\mathbb{R}^3$ question

So the question says: consider the subset $W$ of $\mathbb{R}^3$ consisting of all vectors $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \qquad \text{such that } x+y+z \geq -1 $$ Select all statements ...
1
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1answer
64 views

Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$

Let $V$ be a vector space over a field $k$ and let $U$, $W$ be finite-dimensional subspaces of $V$. Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$ I'm given that to begin this ...
2
votes
1answer
15 views

Applying the Dimensional Formula to Prove a Corollary

I am to prove the following corollary. Let $V_1$ and $V_2$ be subspaces of a $n$-dimensional vector space $V$. If the sum of the dimensions of $V_1$ and $V_2$ is greater than $n$, then $V_1$ and ...
0
votes
3answers
54 views

If $\phi(v_1),…\phi(v_\rho)$ are linearly independent, show that $v_1,…,v_\rho$ are linearly independent

Let $\phi:V\rightarrow W$ be linear. Suppose that $v_1,...,v_\rho \in V$ are such that $\phi(v_1),...\phi(v_\rho)$ are linearly independent in $W$. Show that $v_1,...,v_\rho$ are linearly independent. ...
0
votes
1answer
17 views

Idea behind the tangential vector space?

I am currently reading a chapter about Pfaff forms, but not really understand, why the author introduces tangential vector spaces - the definition seems rather redundant to me, if I didn't overlook ...
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0answers
25 views

Calculate Rotation and Translation Matrix to align elements of input matrix A to Target matrix B in 2d?

I have a matrix in 2D space; the matrix contains elements which I would like to translate into the center of the matrix. Then, I would like to rotate these elements (I mean the positions of the ...
0
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0answers
12 views

Understanding if the definition of constant normal set depends on the choice of the scalar product or not

Suppose we have a Lie group on $\mathbb R^n$, let's say $(\mathbb R^n,*)$. Suppose also that its Lie algebra $\mathfrak g$ is stratified: I mean that there exists a decomposition of $\mathfrak g$ as ...
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1answer
21 views

Why $Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$ is a closed subspace of codimension $1$

Suppose $C^1([0,1]^n)$ is the set of real-valued functions defined on $[0,1]^n$, whose derivative $\leq 1$ is continuous on $[0,1]^n$. Define $$Y=\{ f \in C^1([0,1]^n) : f(0)=0 \}$$ Why $Y$ is a ...
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0answers
25 views

freeness of vector spaces and abelian groups

This question is continuation of my previous question Extension of vector spaces and abelian groups Given a diagram of linear transformations of $K$ vector spaces $$B\xrightarrow{\epsilon} ...
1
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1answer
14 views

Odd Vector Product Question

Here is a question that has me stumped: Use the geometric definition to find: $2 {\bf i} × ({\bf i}+{\bf j})$ Student solution manual says: By the definition of cross product, $2 {\bf i} × ({\bf ...
0
votes
1answer
13 views

Prove a transformation is injective if its restrictions are injective.

Suppose that $V$ is a vector space, and let $V → W$ be a linear map. $$V_0 ⊆ V_1 ⊆ · · · ⊆ V_i ⊆ V_{i+1} ⊆ · · · ⊆ V$$ are subspaces of $V$ (one for each $i = 0, 1, 2, \ldots$) and inclusions ...
0
votes
1answer
40 views

How to show that the axiom for vector space hold for the following operation?

So the operation is sum defined by $f+'g=f\circ g$ (composite of functions) and usual scalar multiplication. First, for $(x+y)+z=x+(y+z)$ property, $(f+'g)+'h=(f \circ g)\circ h\ne f \circ (g\circ ...
0
votes
1answer
35 views

Why is the axiom for vector space not satisfied by the following equation?

Vector sum $(x_1, x_2)+'(y_1, y_2)=(x_1+2y_1, 3x_2-2y_2)$ and the usual scalar multiplication $c(x_1, x_2)=(cx_1, cx_2)$. Sure additive properties does not hold for the operation but why does the ...
2
votes
2answers
33 views

Extension of vector spaces and abelian groups

While reading about modules from Hilton & Stammbach's Homological algebra, I saw the following statement : $\Lambda$ is a ring. $\Lambda$ modules are generalizations of vector spaces and ...
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votes
1answer
26 views

Prove that $v$ is a linear combination of $v_1,…,v_n$ if it is a linear combination of $w_1,…,w_m$

How should I do this? Should I use the definition of linear independence?
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1answer
181 views
+100

Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n $ be proper subspaces of $ \mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
0
votes
1answer
24 views

Vector subspaces of complementary dimensions are not complementary subspaces

If I have a vector space $U$ of finite dimension and two subspaces $V$ and $W$ such that $\dim(U)=\dim(V)+\dim(W)$, then it is not necessarily true that $U=V\oplus W$, right? For example, if in ...
0
votes
1answer
28 views

Cosine Similarity Implementation

This is my first time posting a question in here so please bear with me. I am trying to calculate cosine similarity for two set of data. The cosine similarity formula as I understand is as given ...
4
votes
1answer
51 views

On linear dependence of four vectors in a space of dimention 2?

Let $V$ be a vector space of dimention 2 and $\{v_1,v_2,v_3,v_4\}$ are any four vectors in $V.$ Then we always can find constants $\{c_1,c_2,c_3,c_4\}$ not all zero such that ...
0
votes
1answer
23 views

Rotation matrix according vector

I am stuck on the following two questions. I find formulas for the computation of 3D rotation matrix, but still cannot get how to do those questions. Find matrix for rotation $R_{\theta \bar n}$, ...
0
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0answers
20 views

How to “rotate” on $\mathbb{R}^n$ to maximize minimum pairwise “angle” with set of known vectors?

Say I have $n$ vectors $\{{\bf v}_1,\cdots,{\bf v}_n\}$ which are of unit length: $\|{\bf v}_k\|_2 = 1$ and I want to find new vectors $\{{\bf w}_1,\cdots,{\bf w}_n\}$ by "rotation" i.e. some ...
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votes
2answers
62 views

Why does the associative property of vector addition imply a sum may be written as $\alpha_1+\alpha_2+\cdots+\alpha_n$?

In an effort to understand that a sum involving a number of vectors is independent of the way in which these vectors are associated, I've tried to derive other bindings of certain vector additions in ...
2
votes
1answer
25 views

How to find basis for subspace of matrices

Can someone please guide me with this, I want to find a basis and its dimension for the subspace of 2x2 matrices such that $$\begin{pmatrix} 1 & 2 \\ 2 & 4 \\ \end{pmatrix} \bullet ...
0
votes
0answers
27 views

tensor identity for cross product

I've read somewhere the following identity for a tensor rank 2 $ \nabla \times \nabla v =0 $ where $v$ is a vector of "j" components and $\nabla = \frac{\partial}{\partial x_i}$, such that $ \nabla ...
0
votes
3answers
37 views

Show that set of all $2 \times 2$ matrices forms a vector space of dimension $4$

I have this question: Show that the set of all $2 \times 2$ matrices with real coefficients forms a linear space over $\Bbb R$ of dimension $4$. I know that the set of the matrices will ...
1
vote
3answers
63 views

Why a real Vector Space has either one or infinitely many vectors? [closed]

I have a hard time understanding the above question, so could someone help?
2
votes
0answers
38 views

Relationship between intuitionistic logic and infinite dimensional vector spaces.

Some time ago, I've heard that there was a relationship between intuitionistic logic and infinite dimensional vector spaces. More precisely, the fact that $\neg \neg \phi \to \phi$ may not be "true" ...