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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3
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2answers
198 views

Hahn-Banach Theorem for separable spaces without Zorn's Lemma

I was reading about the Hahn-Banach Theorem, its many versions and their proofs. It's known that in the proofs we need Zorn's Lemma. But in the book that I'm reading, the author said if $X$ is a ...
2
votes
3answers
34 views

Is $n−m$ always the largest possible number of linearly independent vectors in this vector space?

Fix linear independent vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the vector space of such that $S:=$ {$x∈ \mathbb{R}^n:a_i⋅x=0∀1≤i≤m$} . The vector space $S$ always has at least $n ...
0
votes
2answers
31 views

Rewriting a k-form as a wedge product with a 1-form

I am trying to show that a general element of the kth exterior product $\Lambda^kV^*$ (of V an n-dimensional vector space) $$ \alpha = \sum_{i} \alpha_i e_i$$ (where the $\{e_i\}$, for $1\leq i\leq ...
0
votes
2answers
44 views

Why does equation $a · x = 0$ always has $n − 1$ linearly independent solutions for $x$ and never has $n$ linearly independent solutions?

For any nonzero vector $a ∈ \mathbb{R}^n$, why is it that the equation $a · x = 0$ always has $n − 1$ linearly independent solutions for $x$ and never has $n$ linearly independent solutions? My ...
0
votes
1answer
75 views

Why can't vectors $a_1, . . . , a_m$ be linearly independent in $\mathbb{R}^n$ if $m > n$

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the vector space of such that $S:=$ {$x∈ \mathbb{R}^n:a_i⋅x=0∀1≤i≤m$} . Vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$ are linearly ...
0
votes
1answer
22 views

difference between vector space and fundamental matrix - ODE

The solution(s) of a system of first order differential equations seem to be contained in a vector space and as well in a fundamental matrix in the form of columns. Could someone please explain a ...
0
votes
1answer
40 views

Can a subspace of a finite-dimensional vector space contain an infinitely ascending (or descending) chain of subgroups

If $V$ is a finite dimensional $K$-vector space, then every set of subsets of subspaces contains a maximal element, i.e. a subspace which no subspace of the set contains properly, equivalently we have ...
0
votes
0answers
36 views

Fubini's theorem for finite dimensional vector space?

In Weil's Basic Number Theory, the author used Fubini's theorem to show that finite dimensional subspace over locally compact field $F$ has measure zero. While I know this result, I don't know how ...
1
vote
1answer
23 views

$L_1+L_2$ is close if $L_1\bot L_2$ are close sub-spaces of a Hilbert space $H$

$L_1+L_2$ is close if $L_1\bot L_2$ are sub-spaces of a Hilbert space $H$. While I do understand why it is true, I can't be completely sure how deduction is done here. I do know that if $\langle ...
0
votes
1answer
26 views

$A \in M(n,\mathbb R)$ diagonal matrix with charac. polynomial $(x-a)^p(x-b)^q$ , what is the dimension of the space of matrices commuting with $A$?

Let $A$ be a $n \times n $ diagonal matrix with real entries with characteristic polynomial $(x-a)^p(x-b)^q$ , where $a,b$ are distinct real numbers ; let $V:=\{B \in M(n,\mathbb R):AB=BA\}$ then $V$ ...
0
votes
3answers
22 views

How to show $S$ is a vector space?

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the set of $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$. I want to show that $S$ is a vector space. A vector space $S$ is a ...
1
vote
1answer
25 views

Represent standard inner product with respect to a basis

Represent the standard inner product in $R^3$ with respect to the basis ${(1,0,1),(0,1,1), (1,-2,3)}.$ This question is from the book Elements of Differential Geometry. I don not really understand ...
0
votes
2answers
35 views

How to get coordinates with a new basis?

I have this exercise. Consider the following vectors of ℝ²: $$ u_1 = \begin{pmatrix}1\\2\end{pmatrix}, \space\space\space u_2 = \begin{pmatrix}2\\1\end{pmatrix} $$ Determine the ...
0
votes
1answer
18 views

Calculate the coordinates with respect to the basis B given the coordenates of that vector with respect to another basis

first time I post. I can't solve an exercise that I know it's easy, It's so frustrating... First of all: Consider the basis $B_1 = $ {$u_1, u_2, u_3$} in $\mathbb R^3$. 1) Prove that the set $B_2 = ...
1
vote
2answers
23 views

Why, when $m < n$, does the vector space $S$ of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ always contains a nonzero vector?

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let S be the set of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$. Now I know that $S$ is a vector space. Why is it that when $m ...
0
votes
0answers
17 views

Vector spaces/subspaces/basis interaction between them

I have a question resulting from self study,have to let you know that am not a Mathematician so please forgive my ignorance on linear algebra... {Vspace =}\begin{bmatrix} 0 &0 &0 \\ 0 ...
0
votes
1answer
22 views

From $||\alpha'||\geqq\alpha'\cdot\mathbf u$, deduce $L(\alpha)\geqq d(\mathbf {p,q})$, where $L(\alpha)$ is the length of $\alpha$

Let $\alpha: [a,b]\to\Bbb R^3$ be an arbitrary curve segment from $\mathbf p=\alpha(a)$ to $\mathbf q=\alpha(b)$. Let $\mathbf {u=\frac{q-p}{||q-p||}}$, the unit vector from $\mathbf p$ to $\mathbf ...
0
votes
2answers
63 views

Subspaces of $\{f:\mathbb{R} \rightarrow \mathbb{R} \mid f \text{ continuous}\}$

I'm having a hard time grasping vector spaces and subspaces. I'm trying to solve these questions and I know there are axioms to satisfy but I am unsure how to put them into effect. Show that ...
0
votes
0answers
14 views

Find an equation of the plane that passes 2 points and parallel to the intersection of the two planes

For example, I have two points on the plane A,B. And I can get the vector V parallel to the intersection of two planes by using cross product of two normal vector. Then I need to find the normal ...
0
votes
0answers
17 views

extracting the base of a subspace without any knowledge of it

I would like some help with some basic concepts on linear algebra... Thanks in advance! Vspace = ...
0
votes
0answers
35 views

Characterisation of ideals of the endomorphism ring of a vector space.

$V$ - finite vector space over field $F$. Proof, that for any left ideal $I$ of algebra $End_F(V)$ exist only one subspace $W$ of space $V$, which $I = \{A \in End_F(V) | W \subseteq KerA\}$ UPD: ...
0
votes
2answers
34 views

Proving additive inverse of vector set exists and “works”

Let V = {$a_1, a_2): a_1, a_2 \in F$} where F is a field. Define addition of elements of V coordinate wise, and for $c \in F$ and $(a_1, a_2 \in V$}, define $c(a_1, a_2) = (a_1, 0)$. In my proof, I ...
0
votes
0answers
36 views
1
vote
1answer
100 views

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
0
votes
1answer
35 views

Finding all subspaces of $F_2^2$

Let $F_2$ be the field with $2$ elements. List all subspaces of $F_2^2$ and prove the list is complete. So, we have the vectors $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. So we have maximal $4$ ...
1
vote
1answer
29 views

Proving Linear Independent Vector Space

Let $Z$ be a linearly independent subset of a vector space $D$. Prove that if $W$ $\subseteq$ $Z$ then $W$ is also linearly independent. What I tried: I tried to use the fact that $\alpha_1 z_1 + ...
3
votes
2answers
80 views

Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?

I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, ...
1
vote
1answer
29 views

Find a component of a vector orthogonal to two vectors

$$\mathbf u = \begin{pmatrix} 2 \\ 14 \\ -4 \\ 1 \end{pmatrix},\mathbf{v_1} = \begin{pmatrix} 1/\sqrt{5} \\ 2/\sqrt{5} \\ 0 \\ 0 \end{pmatrix}, \mathbf{v_2} = \begin{pmatrix} 2/\sqrt{30} \\ ...
2
votes
1answer
37 views

What is the precise relationship between vector space and vecctor field?

I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them. On wikipedia A vector field is: Given a subset $S$ in $R^n$, a ...
0
votes
1answer
22 views

When working with unit vectors, do we consider the scallor part?

I want to know for perhaps computing dot products etc, that if Im just told the angle between to unit vectors...say pi/6, how would I find the dot product of these two vectors?
1
vote
3answers
31 views

What does this mean in $R^3$ 2x-y=0

Is this a line or a plane, I thought it would be a plane where z=0 always so it will be the xy plane. Also: what will be the normal vector for this if it is a plane?
0
votes
1answer
41 views

Reference request: Proof that every product of vector space is isomorphic to the tangent bundle

On Wikipedia, it says On every tangent bundle $TM$, considered as a manifold itself, one can define a canonical vector field $V : TM → TTM$ as the diagonal map on the tangent space at each ...
0
votes
2answers
22 views

Find a basis of a subspace $S=\{(x_1,x_2,x_3,x_4,x_5)\in\mathbb{R^5}|x_1=x_3=x_5,x_2-x_4=2x_1-x_3\}$

Let $S=\{(x_1,x_2,x_3,x_4,x_5)\in\mathbb{R^5}|x_1=x_3=x_5,x_2-x_4=2x_1-x_3\}$ is a subspace of $\mathbb{R^5}$. Find a basis of $S$. Expand a basis to a basis of $\mathbb{R^5}$. Question: How to find ...
0
votes
1answer
21 views

Find reflection in a cube

Let C be a cube in $R^3$, $C=\{(x,y,z): 0\leq x,y,z,\leq 1\}$. Find a reflection of a diagonal of a face with respect to a plane orthogonal to main diagonal. I am trying to study Vector Calculus by ...
4
votes
3answers
83 views

Linear independence in vector spaces of infinite dimension [closed]

Let $V$ be a vector space which has a countable basis. Any set with an uncountable number of elements will hence have to be linearly dependent. I don't know how to prove the statement above. It ...
0
votes
2answers
42 views

What is the relationship between the relations defining a subspace of a vector space and its dimension?

I was reviewing some linear algebra and in looking at some questions which involve finding a basis for a subspace defined in terms of relations between vector components, I thought about the above ...
0
votes
2answers
28 views

Prove that $(T,+,\cdot)$ is a vector space and find its dimension and one basis

Let $T$ is the set of all $(a,b,c)$ such that the system \begin{cases} 3x+2y+z=a\\[3px] x+y+4z=b\\[3px] 5x+2y-2z=c \end{cases} is consistent. Prove that $(T,+,\cdot)$ is a vector space and find its ...
3
votes
3answers
130 views

Is any subspace of a direct sum necessarily a direct sum of subspaces?

If I have a direct sum $V = V_1 \oplus V_2$ and a subspace $W \subset V$, it it necessarily true that $W = W_1 \oplus W_2$ where $W_1 \subset V_1$ and $W_2 \subset V_2$? I believe this is true ...
-2
votes
2answers
35 views

a vectorspace, a linear map, the kernel and image of it [closed]

I must solve this homework, but I've reached my limits quite fast.. Let $K$ be a field, $V$ a $K$-vector space of finite dimension, and $\Phi∶ V \to V$ a linear transformation. I must prove that ...
2
votes
2answers
29 views

How does the span of vectors [1, 2] and [0,3] equal R2?

I'm watching the video tutorial on spans here: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span At 8:13, he says that the vectors ...
0
votes
0answers
15 views

Decompose $(\mathrm{Sym}^2 \mathbb{C}^2) \otimes (\mathrm{Sym}^2 \mathbb{C}^2)$ into irreducible representations of $\mathrm{SL}_2 \mathbb{C}$

Question: Let $V=\mathbb{C}^2$ be the standard representation of $\mathrm{SL}_{2}\mathbb{C}$. Decompose $(\mathrm{Sym}^2 V)\otimes (\mathrm{Sym}^2 V)$ into irreducible representations $\mathrm{SL}_2 ...
-2
votes
1answer
44 views

Does a basis for $V \otimes_{\mathbb{F}} W$ always consist of pure tensors?

Given a field $\mathbb{F}$ and two $F$-vector spaces $V$ and $W$, it's true that if $\{v_i\}$ and $\{w_j\}$ are bases for $V$ and $W$, respectively, then the set $\{v_i \otimes w_j\}$ is a basis for ...
2
votes
1answer
26 views

$A$ be $n×n$ matrix $A^{n}=0$ ,$A^{n-1}$ not equal to zero a vector $v$ belongs to R^n.then how to proof {V,AV,…A^(n-1)V} is a basis. [duplicate]

Given $A$ be $n×n$ matrix such that $A^{n}=0$, but $A^{n-1}$ not equal to zero a vector $v$ belongs to $\Bbb{R}^{n}$. Proof that {$V,AV,\cdots,A^{(n-1)}V$} is a basis.
0
votes
1answer
47 views

Transpose of the differentiation operator

Please help me write down a step by step solution to the following problem Let $n$ be a positive integer and let $V$ be a finite dimensional vector space of all polynomial functions over the field ...
1
vote
1answer
29 views

Multiple parametric equations for planes and lines $\mathbb R^3$?

I want to know if you can get different sets of parametric equations for a particular line or plane in $\mathbb R^3$? The reason being I know you can have multiple directional vectors or normal ...
1
vote
1answer
24 views

Tangent Surface to a 4D Surface

I have been typing up notes for Multivariable Calculus. While doing so I have been pondering the terms I ought to use for higher dimensional surfaces and the associated tangent surfaces. With a curve ...
3
votes
3answers
129 views

Concerning $f(x_1, \dots , x_n)$

I am not getting even an intuition as how to do this problem. Please help me with a solution.. Let $n$ be a positive integer and $F$ a field. Let $W$ be the set of all vectors $(x_1, \dots , x_n)$ in ...
1
vote
1answer
55 views

Prove that the set of commuting matrices is a vector space

Prove that the set of real commuting matrices with the matrix $A= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ ...
0
votes
2answers
94 views

How to make sense of this linear algebra question about union of proper subspaces

I am having trouble understanding the following; I want to show that a vector space can never be written as the union of two proper subspaces, were proper subspace refers to being a subspace, yet not ...
1
vote
1answer
30 views

$f(\alpha _I) \ne 0$

I need help in this question... Let $F$ be a field of characteristic zero and let $V$ be a finite dimensional vector space over field $F$. If $\alpha _1,\dots , \alpha_m$ are finitely many vectors in ...