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
35 views

Proving that $E=F\oplus G$ for two given subspaces of $E = \mathbb R^3$

Suppose that $F ={(x,y,z)\in \mathbb{R}^3 |x−y+z=0}$ and $g=(1,1,1)$ with $G=Vect(g)$ How can I prove that $E=F\oplus G$? I'm wondering how many ways exist to prove that?
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1answer
16 views

Unitary matrix that diagonalizes real symmetric matrix

I am confused by the following statement from a book: Since a symmetric matrix is Hermitian, it can be diagonalized by a unitary matrix. But since the eigenvalues of a symmetric matrix are ...
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1answer
216 views

Dual operator relationship with complex conjugate.

Let $V$ be a $n$ dimensional vector space spanned by $\{e_{i}\}_{i=1}^{n}$. Let $T:V\to V$ be a linear operator with matrix transformation $A$. Is there any relationship between the dual operator ...
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2answers
57 views

I see some contradiction in the definition of orthogonal vectors

Let's look at a well-known definition of orthogonal vectors: Let $V$ be a vector space. Two vectors $x, y \in V$ are orthogonal to each other when the following condition is fulfilled: $$\langle ...
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1answer
29 views

Coordinate Matrix

I am really struggling with the concept of coordinate vectors and hence coordinate matrix in vector spaces. It would be great if anyone could provide me any intuitive picture to understand it.
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32 views

Proof of “Every vector space has a basis $\implies$ AC” without mentioning von Neumann hierarchy

I am writing a short (30-50 pages) report on AC for an exam. I really would like to include the proof that "Every vector space has a basis $\implies$ AC". Actually, every proof I could find proves ...
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0answers
13 views

Strict convexity and best approximations

Let $V$ be a normed vector space. It is said to be strictly convex if its unit sphere does not contain nontrivial segments. A subset $A \subset V$ is said to have the unicity property if for any $x ...
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2answers
49 views

Proof of the Schwarz's inequality

Let $V$ be a vector space where dot product is defined. Then the following is true: $$\forall x, y\in V \quad \langle x,y\rangle^2 \leq \langle x,x \rangle\langle y,y \rangle$$ Proof: ...
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1answer
17 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
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1answer
8 views

detect that the vector in the any opposite directions to the current

I little bit stuck with quiz - I have vector 'a' and need function which will be checking any other vector, what that vector in negative y'(relatively to vector 'a') I draw simple image below to try ...
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1answer
30 views

Isometry in a finite dimensional vector space is always surjective

My book defines an isometry as a linear operator between two vector spaces X and Y where: $$\|T(x)\|=\|x\|$$ Later it has a sentence which I do not understand. If we have a finite dimensional ...
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1answer
126 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
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1answer
325 views

Can a vector subspace have a unique complement in absence of choice?

Let $V$ be a vector space (not necessarily finite dimensional and over some arbitrary field), and $W$ a proper non-zero subspace. If we assume existence of bases, it is easy to show that $W$ can be ...
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1answer
822 views

Orthogonal Projection of a Point onto a Plane

I'm dealing with an exercise that requires I find the orthogonal projection of a given point onto a given plane. I don't want an answer directly for my exercise, I would instead like to understand ...
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1answer
27 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
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2answers
78 views

The difference between vector space and group

When comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions? ...
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1answer
45 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
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0answers
30 views

Vector (scalar) product: associativity

Let $x$, $y$, $z$ be vectors of $\mathbb{R}^{n\times1}$. Consider this scalar result: $b = x^{\top} y z$. The issue is that the above product does not follow the classical associativity algerbra ...
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1answer
558 views

compute the similarity between two vectors

Euclidean distance is a measure that may be used to compute the similarity between two vectors. Given a query $q$ and documents $d_1, \ldots, d_n$, we may rank the documents $\mathcal{D} = ...
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0answers
36 views

What is the difference between the span of a set to its subspace?

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the ...
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7answers
18k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
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1answer
30 views

Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
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1answer
33 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
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1answer
79 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
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1answer
25 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
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Division of elements of vectors with each other

Suppose we have a vector, like: x = [3,5,7,9,2,3] What does the division of elements from each other, left to right, indicates? Illustrating: ...
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3answers
3k views

how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?

In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof: $$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ ...
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2answers
41 views

Subspace Equations

I'm studying the book "Finite Dimensional Vector Spaces" by Paul Halmos. I'm doing q5 from $\S 12$ Dimension of a Subspace, in chapter $1$. I'm not all that used to L.A. proofs, so I'm looking for ...
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1answer
49 views

linear transformation $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}^4$ which map the following vectors

Is there a linear transformation $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}^4$ which map the following vectors $\begin {pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \mapsto \begin {pmatrix} 1 \\ 1 \\ 1 ...
2
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3answers
318 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
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1answer
21 views

Determining a spanning set for $X/\bigcap_{i=1}^N \ker{\lambda_i}$, where each $\lambda_i$ is a linear functional on $X$

Let $X$ be a vector space over a field $K$. Suppose that $\{\lambda_i\}_{i=1}^N$ is a collection of linear functionals $\lambda_i : X \to K$. Let $W$ be the subspace $\{ x \in X \mid \lambda_i(x) = ...
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4answers
86 views

Suppose that $V_1$ and $V_2$ are subsets of a vector space…

Suppose that $V_1$ and $V_2$ are subsets of a vector space, is $span(V_1\cup V_2) = span(V_1)\cup span(V_2)$? This seems like it should be pretty straight-forward but something is baking my noodle. ...
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0answers
13 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
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2answers
40 views

closeness of a set of vectors

Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$ A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ...
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2answers
79 views

Definition of dimension

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using ...
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0answers
18 views

Is the following subset a vector subspace of F(R,R). How to prove that it is a subspace.

Is the following subset a vector subspace of F(R, R)? The set of functions f : R → R such that f(x + 2) = f(x) for all x ∈ R I know this is obviously a subspace, but the mark scheme didn't ...
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0answers
21 views

Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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1answer
6 views

Find the acute angle made by vector $OC$ and the x-axis.

Given that vector $OA$ = $3i+5j$, $OB$ = $-2i+6j$ and that $OC$ = $OA + OB$, calculate i) |OC|, ii) the acute angle made by vector $OC$ and the x-axis. I found i) $\sqrt122$ Please help me in ...
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3answers
61 views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)1\})$ and determining its dimension.

We consider a space F(R,R) of functions of R in R. Let A = ({1, \sin(x), $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the identity ...
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1answer
56 views

Abelian group over a field underlying an abstract vector space [closed]

Given that a set V is said to be a vector space over a field F if V is an Abelian group under addition and for each $a\in F$ and $\boldsymbol{v}$ in V there is an element $a\boldsymbol{v}$ in V, how ...
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3answers
46 views

What is the cardinal of a field F_5 vector space of dimension 3?

What is the cardinal of a field F_5 vector space of dimension 3? The mark scheme says since F_5 = { 0,1,2,3,4 } there are 5 possibilities. so it is 5^3. So the card(v) = 125. But in the lecture ...
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2answers
23 views

About the subspace of polynomial vector space

Why the set of functions in $C\left [ 1,-1 \right ]$ such that $f\left ( -1 \right )= f\left ( 1 \right )$ is the subspace of $C\left [ 1,-1 \right ]$?
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1answer
13 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
2
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1answer
49 views

Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
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2answers
33 views

groups products vs vector space products

I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted ...
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0answers
18 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
1
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1answer
42 views

Orthogonal Operator Infinite Dimensional Inner Product Space

I know that on a finite dimensional inner product space, a unitary (or orthogonal) operator preserves the inner product. That is, having $\|T(x)\|=\|x\|$ for all $x\in V$ is equivalent to having ...
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3answers
27 views

Determine if $S$ a vector space when it is a subset of $C([-1,1])$.

Let $C([-1,1]$ be the set of continuous functions on the interval $[-1,1]$, and let $S$ be the subset of $C([-1,1])$ consisting of $f$ such that $f(-x)=-f(x)$ for any $x$ in $[-1,1]$. Is $S$ a vector ...
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2answers
24 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
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1answer
215 views

Why is the vector chosen by the right hand rule?

I'm reading first year Physics and the Young & Freedman (13e) text describes how to find the vector (cross) product. Notably, the authors simplify the description of finding the product direction ...