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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Is there a counter example to disprove the following regarding vector addition in binary field?
Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
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The $8$ laws of the definition of vector spaces are not independent. What is the relationship among them? [duplicate]
I am learning linear algebra, and have a question about the eight laws in the definition of a vector space.
A non-empty set $V$ is a vector space on field $F$, if addition $+:V\times V\to V$ and ...
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
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What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
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Applying vector decomposition multiple times and RH orthonormal bases
I want to show that I can write any 3D vector $v$ in components with respect to the right handed orthonormal basis $\{e_1, e_2, e_3\}$ (i.e. three perpendicular unit vectors $\{e_1, e_2, e_3\}$ such ...
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Constructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
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Is this a valid vector space? (Question Verification) [closed]
Hi - please see the question above. I have a problem with proving associativity i.e. that $(\alpha \beta)v = \alpha(\beta v)$ where $\alpha, \beta \in \mathbb{R}$ and $v \in \mathbb{R}_+$.
I think ...
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Distance between subspaces after premultiplication by diagonal matrice
Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be two k dimensional subspaces in $\mathbb{R}^n$ and two $n\times K$ matrices $E_1$,$E_2$ are basis matrix. Distance between two subspaces is defined to be the ...
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Orthogonal complement with respect to a subspace, and then with respect to the larger space.
Suppose I have the subspaces $W\leq V \leq \mathbb{F}_q^n$, with $n$ finite. Let $\langle ,\rangle\colon\mathbb{F}_q^n\times \mathbb{F}_q^n \rightarrow \mathbb{F}_q$ be the dot-product. If I then take ...
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Bilinear forms and reflexivity [duplicate]
Let $V$ be a finite dimensional vector space over a field $\mathbb{K}$, and let $\varphi: V \times V \to \mathbb{K}$ be a bilinear form on $V$. Let's give some definitions. We say that $\varphi$ is ...
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Tensor Product of a vector space with itself.
I've been reading up on tensor products and have been coming up blank on how to think about $V \otimes_F V$, where $V$ is a vector space over a field $F$.
I only care about what is happening when $\...
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Little trouble understanding a uniqueness proof?
I'm reading Postnikov's Analytic Geometry. Here:
What is happening in there? I understand they may be somehow showing that $x$ is unique but I don't understand what is happening in the equations.
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Does $M_{2,3}(\mathbb{R})$ define a vector space adequately?
I was reading my instructor's notes on vector spaces. I came across the following:
Example 9.2.2 Let $M_{2,3}(\mathbb{R})$ be the set of all $2\times3$ matrices over $\mathbb R$. Show that $M_{2,3}(\...
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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How to find basis of vector fields?
I'm figuring out definition of vector fields over a manifold as differentiations of algebra $C^\infty(M)$ of functions on $M$. How can we find their basis starting from this very definition? I know, ...
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Proving the Set of Periodic Functions with Restrictions Form a Vector Space
I understand that a set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ cannot be a vector space because the set is not closed under the sum of the functions, as discussed here. However, I ...
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Graphical Intuition of a Linear Transformation in terms of Row Vectors
The graphical intuition of a linear transformation (matrix) $A \in \mathbb{R}^{m \times n}$ applied on a vector $\textbf{v}$ in terms of the column vectors $\textbf{c}_i$ of $A$ is quite clear to me:
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Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
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Prove that every two lines in space are equal or disjoint or interact at one point only
The question: Let $V$ be a Vactor space over $\mathbb{F}$, Let $\overrightarrow{v},\overrightarrow{w} \in V : \overrightarrow{v} \neq 0 $ . Then we define
$$L_{w,v}=\{\overrightarrow{w}+\...
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Understanding Equivalence of Matrix Elements in Different Bases for Hermitian Operators
Suppose $Q$ and $R$ are two system (which are represented by state vectors in the vector space V) on the same vector space $V$
$|i\rangle$ is an ortonormal base of $V$
$|i_R\rangle$ is an ortonormal ...
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Motivation behind the defination of scalar multiplication of a Vectorspace over a field
In school, we studied physical notations, such as forces, velocities, and accelerations involving both magnitude and direction. We also called any such entity involving magnitude and direction a "...
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Why is the inner product space defined separately?
While learning about the inner product space, I became curious
why it is defined separately?
In my opinion, there seems to be no difference between defining the inner product space separately and ...
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Show that polynomials with a given factor form a subspace
I have a question for one of my assignments but I don't understand how to solve it.
Let $P_n$ be the set of real polynomials of degree at most $n$, show
that
$S=\{p ∈ P_7:x^2+x+4 $ is a factor of $p(...
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Linear Independence without Vandermonde Determinant [closed]
Let $n > 2$ be an integer, $X_1, \ldots,X_n$ be vectors in a vector space, and $\lambda_1, \ldots, \lambda_n$ are nonzero, mutually different scalars.
I want to prove the following implication:
$$
...
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Two definitions of antisymmetrization of a tensor?
I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following:
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Finding Basis for specific Spline Space
Let $S = \{s \in S: s'(a) = s'(b) = 0 \}$ be the spline space that holds all cubic splines with derivate at startpoint (a) and endpoint (b) =0. I want to find a basis for this vector space. I looked ...
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Axler Theorem 5.33: Understanding assumption WLOG
Theorem 5.33 in Axler's book is ($\mathcal{L}(V)$ denotes the set of linear map $V \to V$):
Suppose $\mathbf{F} = \mathbf{R}$ and $V$ is finite-dimensional. Suppose also $T \in \mathcal{L}(V)$ and $b,...
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Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?
I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise:
Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be
a linear ...
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Show that $\mathbb{R}[x]_{\leq 2}$ has a two dimensional subspace contained in the orthogonal complement of the subspace
Let $V=\mathbb{R}[x]_{\leq 2}$, and let $f$ the bilinear form given by $f(p,q)=\int_{-1}^{1} xp(x)q(x)dx.$ Find a basis $B$ of $V$ such that $[f]_B$ is diagonal, and show that $V$ has a subspace $U$ ...
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