Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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Sum of all possible values for $\lambda$.

If $|\vec{a}| = 1$, $|\vec{b}| = 3$, $|\vec{c}| = 4$ and $|\vec{a}-\vec{b}|² + |\vec{b}-\vec{c}|² + |\vec{c}-\vec{a}|² ≤ 12k+λ,$ and if $k$ is prime number and $λ>0$, then what is sum of all values ...
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Linear Algebra - question on the proof of the Replacement Theorem in Friedberg

I'm not quite fully understanding the author's use of induction to prove the replacement theorem in their book Linear Algebra: The induction hypothesis is, as I understand it, that the theorem holds ...
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Generate a uniformly sampled orthonormal matrix that 'rotates' $k$ vectors $x_0 \in \mathcal{R}^{n \times k}$ into $y_0 \in \mathcal{R}^{n \times k}$

We know that orthonormal matrices $H \in \mathcal{R}^{n \times n} $ are rotation matrices. Is there a general method to uniformly generate rotation matrices that can rotate a given set of vector $x_0 ...
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3 answers
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Basis of polynomial vector space with conditions

I understand that the monomial basis proposed in this answer: $\{1,x,x^2,x^3,\ldots,x^n\}$ spans a regular polynomial vector space, but what process would I use to create a basis when there is ...
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-1 votes
1 answer
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Given a subspace $W$ of $V$ and a linearly independent set $A \subseteq W$, is $A$ linearly independent in $V$?

Can you please show that, if a set $A$ is linearly independent in $W$, a subspace of $V$, then it is linearly independent in $V$? Thank you. Here is my attempt. Let $A = \{ a_1, a_2, \cdots , a_n \}$ ...
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2 votes
1 answer
25 views

Checking whether differential operator maps vector space to itself

Given a vector space $V$ that is spanned by: $\{3^t, 2^{-t}\}$, can we conclude that the differential operator $\frac{d}{dt}$ maps $V$ onto itself? It is my understanding that by looking at the basis ...
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  • 143
1 vote
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On the eigenvalue problem

A standard approach to the eigenvalue problem is as follows. We are looking for solutions $\omega$ of the following: $\Omega\vec v=\omega\vec v \tag{1},$ where $\vec v$ is some element of our vector ...
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Does this implies $\mathcal{M}[T]_{\beta_1 \beta_2}$ is similar to the $\mathcal{M}[T]_{\beta_3 \beta_4}$?

$T\in\mathcal{L}(V) $ where $\dim(V) <\infty$ Consider $\beta_1, \beta_2, \beta_3, \beta_4$ four bases of $V$ . Does this implies the matrix $[T]_{\beta_1 }^{\beta_2}$ is similar to the matrix $[T]...
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The space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable

How can one prove that the space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable? I am trying to think what can contradict seperability, but I didn't have any progress.
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uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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1 answer
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Intersection of vector subspaces

Let $V$ be a vector space over a finite field $\mathbb{F}_q$ and $V_i; ~i = 1,2,3$ be three subspaces of $V$ satisfying $V_i \cap V_j = (0)$ for all $i \neq j.$ Can it be proved that $V_1 \cap (V_2 \...
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-2 votes
2 answers
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Finding a spanning set of a null space [closed]

$$ A= \begin{bmatrix} -3& 6 &-1& 1 &-7\\ 1 &-2& 2& 3&-1\\ 2&-4& 5& 8& -4 \end{bmatrix} $$ Please I have a problem finding the spanning set of a null ...
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1 vote
2 answers
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Prove set is a basis of a vector space

Let $\{x,y\} = \{(x_{1}, x_{2}), (y_{1}, y_{2})\}$ be a basis of $K^{2}$. Prove that for a scalar $a \in K$, $a\neq 0$, the set $\{ x+y, ax\} = \{(x_{1}+y_{1}, x_{2}+y_{2}), (ax_{1}, ax_{2})\}$ is ...
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Cohomology of matrix groups acting on vector spaces under change of basis

Let $F$ be a field and let $F^n$ be the vector space of finite dimension $n$ over $F$. Let $G\leq \textrm{GL}_n(F)$. Apply a change of basis to $F^n$, represented by the matrix $C$, and get the group $...
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Nilpotent map and upper triangular matrix

If we have a map $\phi:V\rightarrow V$ on a vector space $V$ that is nilpotent, then there exists a basis $\underline{\mathbf{v}}$ such that the matrix of $\phi$ with respect to basis $\underline{\...
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1 vote
1 answer
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For a representation $(V, p_V)$ by a finite group and $W = \bigoplus \limits_{i = 1}^{n} V$ calculate $\dim(\text{Hom}_G(V,W))$

Let $(V, p_V)$ be a vector space with a representation by a finite group G. Assume further that $V$ is irreducible and $W = \bigoplus \limits_{i = 1}^{n} V$ with the direct-sum representation. Namely: ...
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1 answer
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Prove a Set is a $\operatorname{range}(L)$

Suppose $L:V\to W$ is a linear mapping. Let $B=\{b_1,\ldots ,b_n\}$ be a spanning set for $V$. Prove $C=\{L(b_1),\ldots ,L(b_n)\}$ is a spanning set for $\operatorname{range}(L)$. So I know that ...
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  • 11
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1 answer
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Optimal (in terms of remaining vector lengths) 2-dimensional projection plane of $n$ $d$-dimensional unit vectors

I have a finite number of $n$ unit vectors in $\mathbb{R}^{d}$. I would like to find a two-dimensional projection plane such that each vector has a length larger than 0 in the projection. Moreover, I ...
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3 votes
1 answer
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Why missing the zero polynomial indicate subset is not closed under multiplication? [closed]

My book has the following question: Is this a subspace of $P_{2}: \{ a_{0} + a_{1}x + a_{2}x^2 \mid a_{0} + 2a_{1} + a_{2} = 4\}$? If it is then parametrize its description. My reasoning for it not ...
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1 answer
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Vector Space Axiom "Proof"

Using ONLY the nine other vector space axioms and clearly justifying each step prove that (-a) + a = 0 I got as far as: Since we are allowed to use all other axioms by the existence of V5 (the inverse)...
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0 answers
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Is there any way to define divergence on the surface itself? [closed]

Usually divergence is defined as a limit of a surface integral by volume as the domain of integration shrinks down. Now, the issue is I am thinking of is, is it possible to define divergence on a ...
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1 vote
1 answer
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What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?

In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". ...
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4 votes
2 answers
256 views

Is this a group or a vector space?

Suppose we have $E=\{\mathrm{Car}\}$ with the following operations: $\mathrm{Car}+\mathrm{Car}=\mathrm{Car}$ and $\lambda\cdot \mathrm{Car}=\mathrm{Car}$ for all $\lambda \in \mathbb{R}$. I'm ...
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3 votes
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Is there a unified description of the geometric derivative?

The exterior derivative ($d$ or $\nabla\wedge$) is a very unifying concept, in that it subsumes the gradient of a scalar, the curl of a vector, and the "divergence" of a bivector, thus also ...
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0 answers
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Can I get a solution of an Algebraic Equation with Vector Fields

I would like to get a solution of an Algaibric Equations system, with the use of vector fields. Is it possible please? The idea is to go from a point in space with some directions, and to get the ...
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-2 votes
0 answers
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clarification on subspaces concerning polynomials

Can someone please clarify why quadratic polynomials are not considered a subspace of cubic polynomials? My book gives the definition below: Vector spaces that are not $R^n$ also have subspaces. The ...
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2 votes
2 answers
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Is the component of a vector along another vector also a vector?

I will surmise what I've learned from a couple of Wikipedia articles below: Vector projection: The vector projection of a vector $\vec{a}$ onto a vector $\vec{b}$ (also known as the vector component ...
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-2 votes
0 answers
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Give any set A1 and A2 that is both a subspace of Rn and A1 U A2 is also a subspace of Rn

Provide and set A1 and A2 where they are both a subspace of Rn and where the union of A1 and A2 is also a subspace of Rn
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-3 votes
0 answers
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Let $S:R^2→R^2$ be linear with $S^2=S$. Prove $S= 0, S=id_{R^2}$ or $∃B$, basis of $R^2$ such that $[S]^B_B=\begin{pmatrix}1&0 \\0&0 \end{pmatrix}$ [duplicate]

Can someone provide Hint to Prove this. Let $S: R^2 \rightarrow R^2$ be linear, such that $S^2 = S$. Prove that $S= 0$, $S=id_{R^2}$ or there exists B a basis of $R^2$ such that $[S]^B_B=\begin{...
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1 vote
1 answer
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How to show that the cardinality of the following set is $2^{n-1}$

Let $\mathbb{F}_2^n$ be a $n$-dimensional vector space over the finite field $\mathbb{F}_2$. Here $n>2$. Suppose that we can write $$\mathbb{F}_2^n=A_1\cup A_2,\ A_1\cap A_2=\emptyset,$$ where it ...
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Deterministic approach to construct a special set in a vector space.

Let $\mathbb{F}_2^n$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_2$. I am looking for a deterministic approach to construct a special set $A$ in $\mathbb{F}_2^n$ such that $A$...
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-1 votes
0 answers
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How to prove the following result related to cardinality of a particular set in a vector space? [duplicate]

Let $\mathbb{F}_2^n$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_2=\{0, 1\}$. Let $A\subset \mathbb{F}_2^n$ be its proper subset and $0\in A$. Suppose that there exists a ...
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1 vote
1 answer
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Confused about why having equal coefficents satisfy closure rules of vector spaces

My book says to verify the below quadratic polynomial is a subspace we take the linear combination of 2 members $$M = \{a +ax+ax^2 \mid a\ \in \Bbb{R}\} = \{(1 +x+x^2) \cdot a \mid a \in \mathbb R\}$$ ...
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1 vote
1 answer
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Two distinct lines for which there is no plane that contains both

So if a plane contains a line, does that mean every point of that line is a point on the plane? And if so, would lines that are skewed and perpendicular be an example of the statement?
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2 votes
1 answer
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Need help for understanding a sum of subspaces

I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?! So for subset in order to be subspace, It should ...
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0 votes
0 answers
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A direct sum of Symmetric and Alternating Bilinearforms

Show that the vector space $\text{Bil}(V)$ of all bilinear forms on $V$ can be decomposed in to the direct sum of $\text{Bil}(V)_{\text{sym}} \bigoplus \text{Bil}(V)_{\text{alt}}$, where $\text{Bil}(V)...
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0 answers
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Prove that $D$ is a field. [duplicate]

Here is the question I want to answer: Let $F$ be a field and $D$ an integral domain which is a finite dimensional vector space over $F.$ Prove that $D$ is also a field. Here are my thoughts: Since $F$...
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Calculate normal vectors for each element of a grid in Python

How can I quickly calculate the normal vectors of each mesh of my grid? (grid is defined by the three matrices Mx, My & <...
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0 votes
1 answer
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Kernel and image of a nilpotent linear map

My question is as follows. Let $T : V \to V$ be a linear map of a finite dimensional vector space V. If $ker T = Im T$, then is $dim V$ even and $T^{2} = 0$? I believe this to be false since I have ...
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1 answer
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Is $-\vec{v}$ in vector space axioms mean $-1$ multiplied with $\vec{v}$?

Is $-\mathbf{\vec{v}}$ in vector space axioms mean $-1$ multiplied with $\mathbf{\vec{v}}$? I got this doubt while i am solving the question below. Let $V$ be the set of positive real numbers ...
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0 answers
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Map input from one vector space to another

I am working on a 3d animation project, where I get 3d coordinates of meshes of a face. Using these coordinates I want to move a different 3d face, which is different from the input face with respect ...
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1 vote
1 answer
36 views

Does the minimal polynomial and characteristic polynomial have same roots over F, for a linear operator on vectorspace V over the field F?

Actually my question is that whether the minimal polynomial and the characteristic have the same root over the field of the vectorspace or do they have the same root over any extension field of F. For ...
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1 answer
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Proving a basis of a vector space [closed]

I'm struggling with the following Question: Let $V$ be a vector space and $B=\{v_1,v_2,...,v_n\}$. Prove that $B$ is a basis of $V$ if and only if $\forall i$ s.t $1\leq i \leq n$: $V=Span(v_i)\oplus ...
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1 answer
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What is an example of spanning set with linearly independent vectors that is not a Hamel Basis?

While trying to distinguish the definitions of Hamel and Schauder bases of infinite dimensional vector spaces, I was told that the true definition of a Hamel basis is not "a spanning set with ...
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1 vote
1 answer
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If $T:X\to Y$ is a bijective bounded linear operator and $X$ is Banach then $T$ is invertible

Let $X$ and $Y$ be normed linear spaces and $T:X\to Y$ a bounded linear operator. Notation: $X'$ denote the dual space of $X$, and $T'$ the dual map of $T$. If $W\subset X$, then ${}^\circ W=\{f\in X'...
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1 vote
2 answers
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Is the set of functions from $[0,1]$ to $\mathbb{R}$ i.e. $[0,1]^{\mathbb{R}}$ a vector space?

While the elements of $[0,1]^{\mathbb{R}}$ satisfy properties of vector spaces (commutativity, associativity etc..), I get the feeling that you can find $f,g \in [0,1]^{\mathbb{R}}$ such that for some ...
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-2 votes
0 answers
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Question re: what makes a zero vector for vector space [closed]

Question from the book Hi All, first time poster, I am confused about part c and d of the question. What does the arrow pointing to the Real number symbol mean? The answer is "constant function ...
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How to showing a function is square-integrable in Quantum mechanics

Looking the A-1 section in Cohen-Tannoudji et al Quantum mechanics Vol. 1 book I get lost in the following: $\psi(r)=\lambda_1\psi_1(r)+\lambda_2\psi_2(r) \in F$ In order to show tha $\psi(r)$ is ...
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1 vote
0 answers
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What is the difference between the cartesian product and direct sum of vectors?

My notes give the cartesian product of the sets $X_1, . . . , X_n$ as $$X_1 × · · · × X_n = \{(x_1, . . . , x_n) : x_i ∈ X_i for 1 \le i \le n\}$$ I believe we can think of a vector space $V$ where $V=...
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Clarification in a question regarding vector space and series convergence

The question I am referring to is Hassani's Mathematical Physics Problem 2.18: Using the Schwarz Inequality to show that if $\{\alpha_i\}_{i=1}^{\infty}$ and $\{\beta_i\}_{i=1}^{\infty}$ is in $\...
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