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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$A \in M(n,\mathbb R)$ 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 $V$ ...
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33 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 ...
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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 ...
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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 ...
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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 = ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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2answers
33 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 ...
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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 $4$ ...
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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 ...
3
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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, ...
4
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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 ...
1
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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
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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 ...
1
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1answer
715 views

Cosine Similarity between two sets of vectors?

I have words represented as vectors, and so I can compare two words using the cosine similarity of each word vector. But, now I'd like to extrapolate that and compare two sentences, each being a set ...
0
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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?
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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?
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2answers
21 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 ...
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1answer
25 views

The inverse of a linear operator in a transformation's vector space

Let $\mathbf S$ be some group of vectors. Let $\mathbf V$ be the vector space of all the functions $f:S \to F $ Let $\mathtt g: S \to S$. Let $\mathtt T$ be a linear operator $V\to V$ defined by: ...
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2answers
41 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 ...
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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 ...
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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
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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 ...
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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 ...
0
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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 ...
2
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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 ...
-2
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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 ...
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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
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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.
2
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1answer
30 views

$T^2$ is the identity operator. Then $V$ is the direct sum of of eigenspaces $V^{(1)}$ and $V^{(-1)}$.

$T$ is a linear operator on a finite dimensional vector space $V$. If $T^2$ is the identity operator, how can I prove that V is the direct sum of of eigenspaces $V^{(1)}$ and $V^{(-1)}$?
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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
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1answer
22 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 ...
1
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1answer
54 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 \\ ...
2
votes
1answer
377 views

Need to determine if a subset is a vector space in $\mathbb R^2$

Its been about a year since I have done anything with vector spaces and subsets of vector spaces, and now I've found that I have forgotten a lot of material. I am given this: Suppose $$\mathbf x ...
1
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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 ...
0
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3answers
50 views

Finding the closest point on a plane in $\mathbb R^3$ to the origin given its parametric equations?

I have a plane in $\mathbb R^3$ and I've found its parametric and plane equations. I thought of setting variables in the parametric equations to zero but, I keep getting different values for the ...
2
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0answers
100 views

Given triangle ABC, how to move point B to a certain angle given that its new location lies within the direction of its old altitude.

I have a 2D coordinate system for 3 known points $A$, $B$, $C$. Given that I can only move point $B$, how can I compute for its new coordinate with a certain angle $\theta$ considering that its new ...
1
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1answer
73 views

Can planes move like vectors?

I know that if I have a vector in $R^3$ I can move it around wherever I want, does the same apply to planes? Or are planes constrained to a location? The reason for this questions is that I was given ...
4
votes
1answer
36 views

Triangularization of matrices over algebraically closed field

A friend of mine is studying physics in first semester and for his next assignment, he has to prove the following theorem: Let V be a finite dimensional vector space over an algebraically closed ...
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2answers
35 views

Kernels and Epimorphisms (“Epic Morphisms”) as quotients of vector spaces

Let $f: A \to B$ be a monomorphism of vector spaces. We know that $\text{coker}(f)= B/A$. Is there a similar relationship between $\ker(g)$ and quotients with $M$ and $N$ given an epimorphism $g: M ...
2
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2answers
1k views

Matrix representation of the dual space

Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let ...
0
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0answers
9 views

Sum and intersection of annihilators. [duplicate]

I am a beginner in this course, please give me a detailed step by step solution of this problem. Let $W_1$ and $W_2$ be subspaces of a finite dimensional vector space $V$. If $X$ is a subspace of ...
0
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1answer
13 views

$g(\alpha )=f(\alpha )$

Can anyone help me prove the following? Let $V$ be a finite dimensional vector space over field $F$ and let $W$ be a subspace of $V$. If $f$ is a linear functional on $W$ , prove there is a linear ...
1
vote
1answer
61 views

Pove that the angle between planes in which origin lies is acute if $a_1a_2+b_1b_2+c_1c_2<0$

Suppose we have two planes $$a_1x+b_1y+c_1z+d_1=0$$ and $$a_2x+b_2y+c_2z+d_2=0$$ where $d_1,d_2 >0 \ or \ <0$ then prove that the angle between planes in which origin lies is acute if ...