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
31 views

Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
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2answers
59 views

Verification of whether $S=\{(x,y,z)\in \mathbb{R^3}:x=z^2\}$ is a vector space

Let's say that we have a set $S=\{(x,y,z)\in \mathbb{R^3};x=z^2\}$ To prove that S is a Vector Space I must prove the 8 properties of a Vector Space, such as: $X,Y,Z \in S$ A1) $X+Y=Y+X$ ...
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1answer
43 views

Does All The Functions From A Group To The Complex Numbers Is A Vector Space

Lets there be a group $X$ $\mathbb{C}^x$ is all the functions $X \rightarrow \mathbb{C} $ Addition is defined $f(x)+g(x)=(f+g)(x)$ and scalar multiplication is defined $(\alpha*f)(x)=\alpha*f(x)$ ...
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1answer
47 views

Project a vector onto the intersection of surfaces

I want to project a vector $\vec v$ onto a surface $S$ defined as the intersection of other surfaces. For example, in 5-dimension I have the surface $S(x_1,x_2,x_3,x_4,x_5)=c$, defined by the ...
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2answers
53 views

Orthogonalization of two Vectors [closed]

Given two vectors $v_1$ and $v_2$, which have a given angle $\theta$≠ $$\frac {π}{2}$$, in between; How would one apply a Gram Matrix to define an inner-product, in order to orthogonalize the two ...
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1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
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1answer
36 views

Determine if the following is a subspace and find its smallest possible subspace of $\mathbb{R}^3$

$U_k = \{(x_i)_{1≤i≤n} \in \mathbb{R}^n\ |\ x_k = 0\}$. Is this a vector subspace of $\mathbb{R}^n$? For $n = 3$, what is the smallest vector subspace of $\mathbb{R}^n$ that contains $U_1, U_2, U_3$. ...
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1answer
28 views

Multiplication by scalar axioms for an abelian group.

There is an R vector space where $k ⊙ x := x^k$ , $∀x, y ∈ V, k ∈ R$, I showed that it was abelian. I wanted to show scalar multiplication by using the axioms. $α ⊙ (x ⊕ y) = α ⊙ (xy) = (xy)^α = ...
2
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1answer
36 views

On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
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1answer
18 views

Finding the expression of a projection

Suppose that $\mathbb{R}^3=K\oplus L$ as $K=Vect(k)$ and $L = Vect(l_1,l_2)$ and $k=(1,2,1)$, $l_1=(1,0,-1)$, $l_2=(-2,1,1)$. And Supposing that $q$ is the projection on $K$ in parallel to $L$. The ...
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2answers
38 views

Need to find basis for a subspace of a vector space [closed]

Recall that $\mathbb R$ is a vector space over field $\mathbb Q$ of rational numbers. Let $V$ be a subspace of $\mathbb R$ given by $V=\{a+b\sqrt{3}\::\:a,b ∈ \mathbb{Q} \}$. (a) Find a basis for $V$ ...
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1answer
17 views

How is the zero set of an affine transform a vector subspace?

An affine transform $T$ is defined as $T(x)=a(x)+b$, where $a(x)$ is a linear map, and $b$ is a vector. An affine subspace is defined as the zero set of an affine transform. How is the zero set of ...
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2answers
74 views

Prove that the group $(A,+, ◦) $ is a non-commutative ring

• $A × A → A, (f, g) → f + g$, where $(f + g)(x) = f(x) + g(x)$ for all $x ∈ K$ • $A × A → A, (f, g) → f ◦ g$ where $(f ◦ g)(x) = f(g(x))$ for all $x ∈ K$ Show that $(A,+,◦)$ is a non commutative ...
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3answers
250 views

Counterexample for Mazur–Ulam theorem

We know, Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping $f\colon V\to W$ is a surjective isometry, then $f$ is affine. Can somebody say ...
7
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2answers
294 views

Uncountable Basis?

I was reading up on the difference between countable and uncountable sets, and was wondering if there was a basis of uncountable size. I now know there are, however they all seem to be covering ...
2
votes
1answer
100 views

Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
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1answer
56 views

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist linear map $F:V\to W$ with ker F=ker $T\cap $ ker S

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist a linear map $F:V\to W$ with ker F=ker $T\cap $ ker S, where $V$ and $W$ are different vector spaces? What if $V=W$? The answer is in ...
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1answer
19 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
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2answers
46 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
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1answer
57 views

Is it true that for an inner product space a norm of a vector is defined unambiguously?

Suppose we have some vector space $V$ for which we have defined an inner product $\langle \cdot\rangle$. Thus we have an inner product space. Is it true that $\forall x \in V : \lVert x\rVert := ...
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1answer
57 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
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1answer
19 views

Dimension of $\cal V^-$ if $\cal V$ is a $\Bbb C$ vector space

In Halmo's book 'Finite dimensional vector spaces' there's a question I'm kind of stuck on in chapter 1. $1 (b)$ Every complex vector space $\cal V$ is intimately associated with a real vector space ...
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1answer
17 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 ...
1
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1answer
44 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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2answers
67 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
28 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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0answers
38 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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2answers
53 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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2answers
36 views

The set consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.

The set $S$ consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.
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1answer
22 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
34 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
142 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
31 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 $$ ...
2
votes
2answers
84 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
48 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
37 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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0answers
31 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
32 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
34 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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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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3answers
323 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
22 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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2answers
81 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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1answer
52 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 ...
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0answers
24 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 $P(t) \in V \Leftrightarrow (t − a)^2$ ...
2
votes
1answer
27 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 ...
0
votes
1answer
7 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 ...
1
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3answers
49 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 ...